Monoids with Absorbing Elements and Their Associated Algebras

Home , Absorbing element, Idempotence

arXiv:1603.02093v1 [math.AC] 7 Mar 2016 ood ihasrigeeet n their and elements absorbing with Monoids erur rf r ogrBrenner Holger Dr. Prof. Betreuer: u ragn e otrrds(r e.nat.) rer. (Dr. Doktorgrades des Erlangung zur soitdalgebras associated e aheecsMathematik/Informatik Fachbereichs des e Universit der Osnabr ioeB Simone Dissertation vorgelegt von c,2015 uck, ¨ tOsnabr ¨ at ¨ ottger uck ¨

Introduction

In what follows, K denotes a ﬁeld, P the polynomial algebra K[X v V ], V := 1,...,n , in n 1 v | ∈ { } ≥ commuting variables X ,...,X , and Xν will be shorthand for the monomial Xν1 Xνn . 1 n 1 ··· n One of the main objects of concern in combinatorics, commutative algebra, and algebraic geometry are algebras P/I deﬁned by an ideal I P , or more generally, varieties and schemes associated to I. ⊆ Among the large amount of examples that appear in these areas some of the most important are the following.

(1) The coordinate ring K[ ]= P/I of an aﬃne (or projective) variety An(K) ( Pn−1(K)), V V V ⊆ ⊆ where I = F P (homogeneous) F (v) = 0 for all v . V { ∈ | ∈ V} (2) The Stanley-Reisner algebra K[∆] := P/I∆ associated to a simplicial complex ∆ on V , where I = ( X F ∆) is the so-called Stanley-Reisner ideal of ∆. ∆ v∈F v | 6∈ (3) The toric ring K[C] K[Y ,...,Y ] of a ﬁnite set of monomials C = Y ν1 ,...,Y νn Q ⊆ 1 r { } ⊆ K[Y ,...,Y ], which is given by P/I via π : P K[C], X Y νi , where I = ker π is 1 r C → i 7→ C the so-called toric ideal associated to C.

As a non-commutative example, we give the following from representation theory.

(4) The path algebra K[Q]= P/ID associated to a quiver Q := (V, A), where D is the set of paths in Q and I = (Xp p D) is the so-called path ideal of Q, where P denotes in this case the D | 6∈ polynomial algebra with non-commuting variables X , v V . v ∈ The algebras K[∆] and K[Q] are related to combinatorial objects, namely ∆ and Q, and so is K[C] since it is naturally isomorphic to the monoid algebra KM of the affine monoid M = Nν + + Nν 1 ··· n generated by ν ,...,ν Nr. While K[∆] and K[Q] are monomial algebras (i.e. defined by an ideal 1 n ∈ generated by monomials), the defining ideal IC of K[C] is a graded prime ideal generated by finitely many pure difference binomials ([56, Proposition 7.1.2]), which are polynomials of the form Xν Xµ. − In fact, every monoid algebra is the homomorphic image of a polynomial algebra K[X v W ] for a v | ∈ not necessarily finite set W by an ideal I generated by pure difference binomials ([28, Theorem 7.11]). This also shows that certain varieties are related to combinatorial objects, for instance, toric varieties ([20],[27]).

From a conceptual point of view it seems desirable to have a theory that treats those algebras (or varieties or schemes) related to combinatorial objects within the same framework. Toric face rings are one attempt. Introduced by Stanley ([54]), they unify Stanley-Reisner algebras and affine monoid algebras by mimicking the connection of a monoid to its algebra with a modified operation described below (⋆). The construction of Stanley has been generalized to monoideal complexes ([34], [11, Chapter 7.B]) and has proven to be relevant to such a unification. Another very powerful framework is the theory of binomial ideals. A binomial is a polynomial of the form aXν bXµ with a,b K, and an ideal generated by binomials is a binomial ideal. By definition, − ∈ a pure difference binomial is a binomial with a = b = 1, and a monomial is a binomial with a = 1 and b = 0. In particular, every Stanley-Reisner algebra and monoid algebra can be realized as an algebra defined by a binomial ideal. Algebras, varieties, and schemes related to a binomial ideal constitute an immense quantity of classical and important examples in combinatorics, commuatitve algebra, and 2 algebraic geometry, which justifies the strong interest as well as the large body of literature and active work on binomial ideals ([11], [24], [33], [41], [56], [35]). Moreover, they have notable applications beyond pure mathematics like coding theory, algebraic statistics, game theory, linear programming, and chemical kinetics ([55], [40], and the references therein). Due to their combinatorial nature, a lot of algorithms and feasible computations are possible, which makes them even more attractive.

However, the somewhat smaller class of binomial ideals generated merely by pure difference binomials and/or monomials (i.e. by binomials with coefficients a,b 0, 1 ), which are independent of the base ∈{ } field (or ring), still cover a great part of the significant cases (e.g. toric face rings). This class outlines precisely the scope of this thesis. The main reason for restricting ourselves exactly to this class is that it is the largest class within which the powerful interplay of monoids and their algebras can be extended in a rather simple way.

Definition. Let M be an additively written monoid. An absorbing element a M satisfies a + b = a ∈ for all b M. Such an element is always unique and will be denoted by if it exists. A monoid ∈ ∞ with an absorbing element is called a binoid. Its associated algebra, called the binoid algebra of M, is defined to be the quotient algebra K[M] := KM/(X∞) .

By definition, every binoid is a monoid. On the other hand, adjoining an absorbing element to an arbitrary monoid M subject to its defining property yields a binoid M =: M ∞ such ∪ {∞} that K[M ∞] = KM. Thus, binoid algebras generalize monoid algebras. They are precisely the homomorphic images of polynomial algebras by ideals generated by pure difference binomials and/or monomials. Since adjoining an absorbing element has no impact on the structure of a monoid there is no loss of generality when turning from a monoid to a binoid. As a matter of fact, dealing with binoids is more natural for several reasons. As in ring theory, one of the most fundamental tools in monoid theory is the Rees quotient

M/ = [a] a M [ ] = M [ ] I { | ∈ \I}∪{ I } ∼ \I∪{ I } of a monoid M by an ideal M with addition given by I ⊆ [a + b] , if a + b , [a] + [b] = 6∈ I ([ ] , otherwise. I On the one hand, the class [ ] is an absorbing element in M/ , which shows that one inevitably I I encounters binoids even when dealing solely with monoids that admit no absorbing element in first place. On the other hand, considering M ∞ yields a canonical representative for the class [ ], namely I [ ] = [ ], because it reflects its characteristic feature of being absorbing. By proceeding in this way, I ∞ one obtains an elegant description of M ∞/ , which is also appreciated in monoid theory but has to I be enforced by the standard practice of defining [ ] =: ([30, Chapter 1.3]). I ∞ Considering binoids accomplishes more than just focussing on a particular class of monoids that appear naturally in many situations, and yielding better notation when the absorbing element is emphasized. The main motivation for binoid theory was the observation that many algebras that arise in combinatorics can be realized as binoid algebras but not as monoid algebras, because of zero- divisors that can be encoded on the combinatorial level with binoids but not with monoids. Let us outline how such algebras are typically constructed: given a collection C of objects with a partial operation, that is, to some pairs c ,c C there exists a unique object in C, say c c , and 1 2 ∈ 1 ∗ 2 whenever c c and c c are defined for c ,c ,c C, then (c c ) c is defined if and only if 1 ∗ 2 2 ∗ 3 1 2 3 ∈ 1 ∗ 2 ∗ 3 c (c c ) is as well, and in this case they coincide. The algebra associated to C is now defined to 1 ∗ 2 ∗ 3 Introduction 3 be the K - module K[C] := KXc cM∈C with multiplication defined by

c1∗c2 X , if c1 c2 is deﬁned, Xc1 Xc2 := ∗ (⋆) · (0 , otherwise.

Obviously, if the partial operation is no operation, this algebra is not an integral domain and cannot be realized as a monoid algebra. However, it is straightforward to see that C deﬁnes a binoid, namely C =: M with addition given by ∪ {∞} C

c1 c2 , if this is defined, c1 + c2 := ∗ ( , otherwise, ∞ and that K[MC]= K[C]; that is, K[C] is a binoid algebra. Hence, shifting the defining operation from the module to the combinatorial level creates a situation similar to the correspondence of monoids to their algebras. This is to say, with a theory of binoids and their algebras at hand, the strucure of K[C] could be revealed by the combinatorics of MC and vice versa. The advantages of such a theory are obvious. Like toric face rings, binoid algebras unify different algebras that arise in combinatorial commutative algebra by studying them within the same framework. This is a very powerful and comprehensive framework since binoids being monoids allow us to adapt the well-established and elaborate theory of monoids and their algebras, given for instance in [15], [28], [29], [30], [38] or [46] to mention just a very few (of which some also deal with binoids and their algebras like the books [15] from the 1960s). Of course, considering the absorbing element causes new properties to arise, known concepts need to be modified, and pathologies appear compared to common structures in (commutative) algebra. However, for several reasons and motivations, working with binoids rather than arbitrary monoids seems preferable in certain situations, as can be verified by the frequency with which they appear in the literature and their profound applications in different branches of algebra and geometry ([3], [15], [30], [37], [39], [43], [44], [46]). There, they are usually written multiplicatively and called “monoid with zero” or “pointed monoid”, while the binoid algebra is also known as a “contracted” or “pointed monoid algebra”. For instance, spaces with a monoidal structure became increasingly important due to their appearance in F1 - geometry and logarithmic algebraic geometry within the last two decades. Study in both areas depends largely on the commutative algebra of monoids and consequently enriches monoid theory by developing a geometric theory of monoids including sheaves of monoids and monoid schemes ([45], [21], [22]), with which we will not be concerned. Even there, some approaches favor working with binoids from scratch for a more general theory ([14], [16], [17], [18], [19], [26]). Thus, binoids appear in many different branches of mathematics but, unfortunately, it seems that one is not always aware of what has already been done elsewhere. This is most likely due to terminology, since in older works on binoids, they are simply addressed as semigroups.

We also want to bring two more works on binoids to the attention of the reader. In [6], Hilbert-Kunz theory for binoids is developed. It is shown that for certain binoids the Hilbert-Kunz multiplicity exists and is a rational number. A geometrical approach to binoids can be found in [1], where invariants of a binoid scheme are related to those of the corresponding ring. The main focus of this work are divisor class groups and Picard groups of binoid schemes.

It is one aim of this thesis to provide a general theory of binoids, their algebras, and their spectra, but we will not pay as much attention to the algebras (and modules) as to the binoids and spectra. 4

The core of this thesis is rather based on the familiar idea in commutative algebra to go one step further and have a view on algebraic geometry by means of the seminal relation between commutative algebras and geometric spaces. In Chapter 4 and 5, we show that the space of K - points of a binoid algebra can be studied and fully understood from the combinatorics provided by the underlying binoid by making use of the crucial isomorphism

K-spec K[M] ∼= K-spec M, where K-spec M is the set of all K - valued morphisms of M. In this regard, we investigate several connectedness properties of such spaces. In the last chapter, we apply binoid theory to simplicial complexes, determine those binoids that realize Stanley-Reisner algebras, and show how a theory of these objects could start from a binoid point of view.

This thesis contains six chapters, which we will sketch now. The first two chapters cover binoid theory including the ideal theory of binoids to which we return later when we investigate the spectrum of a binoid from a topological point of view. Within the scope of this thesis, it was not possible to give a full account of the whole of binoid theory developed so far and elsewhere. We try to compensate this by giving references but do not claim completeness. More precisely, Chapter 1 is a collection of basic definitions and constructions for binoids. Here, terminology and mainly additive notation are taken from monoid theory. The main definitions and properties of binoids are given before we focus on N -spectra of binoids, where N is another binoid. In view of our main concern, the N - spectra are determined for every construction dicussed in this chapter. After recalling the well-known concept of congruences for a binoid, we collect specific congruences of interest for us. We prove that the product, coproduct, and limits exist in the category of (commutative) binoids, and describe these in detail. For binoids (resp. pointed sets), one can also define the pointed and bipointed union, where the latter applies only to binoids with a trivial unit group. The module and algebra counterparts in binoid theory are described by pointed sets and binoids over a binoid. We close this chapter with considerations on localization and integrality properties including the normalization of a binoid. Though we are mainly interested in finitely generated commutative binoids and devote one section to them, the first chapter is kept as general as possible, whereas in the remainder, we deal almost exclusively with commutative binoids. In Chapter 2, we investigate the ideal theory of commutative binoids with focus on the spectrum of a binoid, which corresponds one-to-one to its 0, - spectrum and to its filtrum, which is the set { ∞} of all filters. The spectra of several binoid constructions are determined by taking advantage of our knowledge of their N - spectra. The ideal theory of binoids is very similar to that of rings with the exception that the union of (prime) ideals in a binoid is again a (prime) ideal. Moreover, every binoid can be considered to be local in the sense that there is a unique maximal ideal with respect to set inclusion. In Chapter 3, we move on to modules and algebras associated to binoids, N - sets, and N - binoids. After recalling useful results on monoid algebras, we introduce binoid algebras, state basic facts about them, and investigate the relationship between their ideals and the ideals of the underlying binoid. Modules and (binoid) algebras over binoid algebras generalize the concept of binoid algebras and will be treated at the end of this chapter. In Chapter 4, we take a topological approach to commutative binoids via their spectra and K - spectra, where K denotes a field. The spectrum of a binoid can be endowed with the Zariski topology in the same manner as that of a ring and the theories are very similar. This also applies to the dimension theory of binoids, which we treat in the third section. We introduce the booleanization of a binoid, whose spectrum is homeomorphic to that of the initial binoid, but which is a fairly easier binoid to study. The topology on the K -spectrum of a binoid comes from that on the K -spectrum of the binoid algebra K[M]. We prove some basic facts before we investigate connectedness properties of Introduction 5 the K -spectrum of K[M] that can be analyzed solely on the combinatorial level. Here, we focus in particular on the case of hypersurfaces. In Chapter 5, we study a stronger connectedness property which involves the (special) K - point that is related to the unique maximal ideal in the binoid. We give necessary and sufficient conditions under which the special point is contained in every cancellative component of the K - spectrum, and under which every K - point is An - connected to the special point. To obtain these results, we develop the notion of separated and graded binoids. In Chapter 6, simplicial complexes and Stanley-Reisner algebras are studied from a binoid theoretic point of view. After recalling basic facts on simplicial complexes, we consider two binoids defined by a simplicial complex (with respect to and ). The first three sections deal with the basic properties ∪ ∩ of these binoids, their spectra, and morphisms between them. For a binoid N, the N - points of a simplicial complex ∆ are introduced. These correspond one-to-one to the N - points of the binoid M∆ associated to ∆, which is the main concern of the last section. We prove that the binoid algebra of

M∆ is given by the Stanley-Reisner algebra and characterize those binoids that realize Stanley-Reisner algebras.

Conventions

Throughout this thesis we make the following agreements. The sum over the empty set is 0, and the product over the emptyset is 1. By a ring, we always mean a commutative ring with identity and all ring homomorphisms preserve the identity.

Acknowledgments

It is a great pleasure to thank all of those who helped me along my way. My deepest thanks go to Holger Brenner whose expertise, understanding, and enthusiasm added considerably to my postgraduate experience. I would like to thank him for all his support during the past years and for an excellent supervision of this thesis. I always enjoyed working with him and beneﬁtted a lot from numerous interesting discussions. I take this oppportunity to express my gratitude to my past teacher Uwe Storch who introduced me to commutative algebra. He allowed me to work with him and to share some beautiful piece of mathematics. I also owe him the chance to spend six months of graduate years at the Department of Mathematics of the Indian Insitute of Science, Bangalore, India. I would also like to thank Lukas Katth¨an, Davide Alberelli, Sean Tilson, and Katja Brunkhorst for suggestions and proof-reading parts of this thesis.

Contents

Introduction 1

1 Basic concepts of binoids 9 1.1 Basicobjects ...... 9 1.2 Homomorphisms ...... 12 1.3 Generators ...... 13 1.4 Additivity properties ...... 17 1.5 N -spectra...... 25 1.6 Congruences...... 28 1.7 Importantcongruences...... 31 1.8 Smashproduct ...... 35 1.9 Pointedunions ...... 40 1.10Operations ...... 44 1.11 Direct and projective limits ...... 51 1.12 Finitely generated commutative binoids ...... 54 1.13Localization...... 68 1.14Integrality...... 73

2 Ideal theory in commutative binoids 77 2.1 Ideals ...... 77 2.2 Primeideals...... 83 2.3 Minimal prime ideals ...... 90

3 Basic concepts of binoid algebras 95 3.1 Monoidalgebras ...... 95 3.2 Binoidalgebras...... 96 3.3 Ideals in binoid algebras ...... 102 3.4 K[N]-modules ...... 103 3.5 Binoid algebras of N -binoids...... 107

4 Topology of commutative binoids 111 4.1 Spectrum ...... 111 4.2 Booleanization ...... 116 4.3 Dimension...... 119 4.4 K -points ...... 121 4.5 Connectedness properties of K -spectra ...... 126

5 Separation and gradings 133 5.1 Separatedbinoids...... 133 5.2 Gradedbinoids ...... 138

6 Binoids deﬁned by a simplicial complex and simplicial binoids 143 6.1 Simplicial complexes ...... 143 8 Contents

6.2 Simplicial morphisms ...... 147 6.3 N - points of a simplicial complex ...... 153 6.4 Topology of simplicial complexes ...... 156 6.5 Simplicial binoids ...... 158

Index 167

Symbols 173 1 Basic concepts of binoids

In this chapter, the basic terminology and notation concerning monoids with an absorbing element, which we call binoids, will be introduced. All definitions come along with examples which will frequently appear in the following chapters. In the remainder of this thesis, we will mainly focus on the class of commutative binoids, but in this chapter we try to keep it as general as possible and consider arbitrary binoids if not otherwise stated. At the beginning of those sections that deal only with commutative binoids, we will mention this restriction. From the beginning, we fix additive notation for arbitrary binoids, though the terminology and notation of classical concepts will not always follow this convention. On the one hand, every binoid is a monoid so that the concepts from monoid theory apply to binoids if the special element (and its inherent property of being a universal absorbing element) has no ∞ effect. In this case, definitions and results of semigroup and monoid theory may be transfered directly to (semi-) binoids, otherwise we will point out the necessity to adapt well-known concepts by making it compatible with . For the sake of completeness, definitions and results are given. Standard ∞ references for semigroup and monoid theory are the books [15] by A.H. Clifford and G.B Preston, as well as [29] and [30] by P.A. Grillet, and [28] by R. Gilmer, where the latter two focus on the commutative case. On the other hand, every binoid can also be considered as a pointed set, where the distinguished point is given by the absorbing element. This concept yields very interesting and useful constructions for binoids, cf. Section 1.8 and Section 1.10.

1.1 Basic objects

In this section, we give the deﬁnitions of the main objects of this thesis, namley binoids and their algebras. We say how to consider a binoid as a pointed set and thereby deﬁne the product and the direct sum of a family of binoids in the same way as for pointed sets.

Definition 1.1.1. A semigroup (M, ) is a set M with an associative operation ∗ : M M M, (a,b) a b . ∗ × −→ 7−→ ∗ A monoid(M, ,e) is a semigroup that admits an element e which satisfies a e = e a = a for ∗ ∗ ∗ all a M. Such an element is always unique and will be called the i d e n t i t y element of M. An ∈ identity element is always unique. A s u b m o n o i d of a monoid M is a subsemigroup that contains the identity element of M. In additive notation, the identity element will be denoted by 0 and in multiplicative notation by 1. An element a M in a semigroup is a b s o r b i n g if a x = x a = a for all x M. An ∈ ∗ ∗ ∈ absorbing element is always unique. A b i n o i d (M, ,e,a) (resp. semibinoid (M, ,a)) is a monoid ∗ ∗ (resp. semigroup) with an absorbing element. A subbinoid (resp. subsemibinoid) of M is a submonoid (resp. subsemigroup) of M that contains the absorbing element of M. In additive notation, the absorbing element will be denoted by and in multiplicative notation by 0. ∞ By definition, semibinoids and monoids are never empty and, in particular, so are binoids. 10 1 Basic concepts of binoids

Convention. In this thesis, arbitrary binoids will be written additively if not otherwise stated (even if the binoid is not commutative).

By abuse of notation, we will not strictly use additive notation when referring to classical concepts such as localization and so forth. Moreover, unless there is confusion, we abbreviate

na := a + + a and nA := a + + a a A ··· { 1 ··· n | i ∈ } for n N, a M, and A M, where 0a := 0 and 0A := .1 ∈ ∈ ⊆ ∅ While we are studying binoids, we always have their associated algebras in mind and will refer to them. Then we tacitly assume little knowledge about monoid algebras since the binoid algebra is given by R[M] := RM/(T ∞) ,

a ∞ where M is a binoid, R a ring, RM = a∈M RT the monoid algebra, and (T ) the ideal in RM generated by T ∞. An elaboration of the basics of monoid and binoid algebras will be given in Section L 3.1 and Section 3.2.

Example 1.1.2. Let R be a ring. (1) The binoid algebra of the zero binoid , i.e. 0 = , over any ring is the zero ring. {∞} ∞ (2) The binoid algebra of the trivial binoid 0, , i.e. with 0 = , over R is R. { ∞} 6 ∞ (3) Adjoining an absorbing element to the commutative monoid (Nn, +, (0,..., 0)), n 1, yields ∞ ≥ a binoid, denoted by (Nn)∞, by deﬁning k + =: (this construction is based on a gen- ∞ ∞ eral concept which will be discussed subsequently). The binoid algebra R[(Nn)∞] = R(Nn) is isomorphic to the polynomial algebra R[X1,...,Xn] in n variables. (4) Every ring yields a (commutative) binoid by forgetting the additive structure.

The binoids of the following example play an important role in this thesis and will frequently appear.

Example 1.1.3. Let V be an arbitraryset. The power set P(V ) gives rise to two different commutative binoids, namely P(V ) := (P(V ), ,V, ) and P(V ) := (P(V ), , , V ) . ∩ ∩ ∅ ∪ ∪ ∅ In either case, P( ) yields the zero binoid and P( 1 ) the trivial binoid. If V is finite, we abbreviate ∅ { } P( 1,,...,n )=:P and write P and P for the corresponding binoids, n 1. { } n n,∩ n,∪ ≥ The subbinoids of P(V ) and P(V ) are given by the subsets M P(V ) that are closed with respect ∩ ∪ ⊆ to the operation of and , respectively, and contain and V . If M is a subbinoid of P(V ) (resp. ∪ ∩ ∅ ∪ P(V )∩), then M c := U c U M , { | ∈ } where U c := V U denotes the complement of U, is a subbinoid of P(V ) (resp. P(V ) ) since \ ∩ ∪ U c W c = (U W )c (resp. U c W c = (U W )c) for all U, W P(V ). ∪ ∩ ∩ ∪ ⊆ In particular, every topology = U U V open on a nonempty set V defines commutative T { | ⊆ } binoids with respect to the join and meet operation, namely ( , ,V, )= and ( , , , V )= , T ∩ ∅ T∩ T ∪ ∅ T∪ as well as the set of all closed sets c = U c U . The binoids P(V ) and P(V ) come from T { | ∈T} ∪ ∩ the discrete topology on V , and the binoid due to the trivial topology on V is nothing else than the trivial binoid V, . { ∅} By definition, every semibinoid is a semigroup and every binoid is a monoid. On the other hand, it is always possible to adjoin an identity or an absorbing element subject to their defining conditions

1 The latter deﬁnitions are consistent with our convention that the sum over the empty set is 0. 1.1 Basic objects 11 even if such an element already exists. For instance, if (M, +, 0, ) is a binoid, then M 0 subject ∞ ∪{ } to a + 0 = 0 + a = a for all a M, in particular, ∈ 0+ 0 = 0 +0=0 , is a binoid with identity element 0. Thus, every semigroup (or monoid or semibinoid) is embeddable in a binoid so that considering binoids amounts to no loss of generality.

Deﬁnition 1.1.4. The semibinoid and monoid that arises from adjoining an absorbing and an identity element to a semigroup M will be denoted by M ∞ and M 0, respectively. If M possesses an absorbing element , we write M • for the set M . ∞ \ {∞} Objects like monoids and semibinoids originate from a more general concept.

Definition 1.1.5. A pointed set (S,p) is a set S with a distinguished element p S and a map ∈ (S,p) (T, q) of pointed sets with p q is called a p o i n t e d m a p . The set of all pointed maps → 7→ S T will be denoted by map (S,T ). In case T = S, we simply write map S. → p7→q p The set map S is a binoid with respect to the composition of maps S S. The identity element is p → given by id and the absorbing element by the constant map c : s p, s S. In the next section, S p 7→ ∈ cf. Lemma 1.2.6, we will show that every binoid can be realized as the subbinoid of such a mapping binoid (map S, , id ,c ) . p ◦ S p Note that there are no conditions required from the distinguished point unless it comes to morphisms of pointed sets. Thus, a set M is a monoid if and only if it is a pointed semigroup (M, 0) with the additional property of the identity element 0. Similarly, a semibinoid M is a pointed semigroup (M, ) subject to the defining property of . Having observed this, a binoid M can be considered ∞ ∞ as a pointed set (M,p) in two different ways: as a pointed set with p = 0 or as a pointed set with p = . It turns out that the latter approach is the one that makes more sense in our context, see for ∞ instance the definition of the product below or Section 1.8 and Section 1.10. Since binoids are monoids, the product and the direct sum of a family of binoids is well-defined. It remains the question if the outcome has a binoid structure as well. For the sake of completeness, we recall the definitions for pointed sets and state the main results about monoids (without giving proofs) before we consider binoids.

Deﬁnition 1.1.6. The p r o d u c t of an arbitrary family (Si,pi)i∈I of pointed sets is given by the cartesian product S . The subset consisting of all tuples (s ) with s = p for almost all i I i∈I i i i∈I i i ∈ is called the d i r e c t s u m and will be denoted by S . Q i∈I i

The product and the direct sum of a family (Si,pLi)i∈I of pointed sets are again pointed sets with distinguished element (pi)i∈I and they coincide if and only if I is ﬁnite; in this case, we prefer the notation i∈I Si instead of i∈I Si. Considering a monoid as a pointed set with distinguished element 0, we get the following result. Q L

Lemma 1.1.7. Let (Mi, +, 0i)i∈I be a family of monoids. The product i∈I (Mi)i∈I is a monoid with the componentwise addition (a ) + (b ) = (a + b ) and identity element (0 ) . The direct i i∈I i i∈I i i i∈I Q i i∈I sum M , which consists of all tuples (a ) with a =0 for almost all i I, is a submonoid of i∈I i i i∈I i i ∈ M . i∈IL i QProof. This is easy to check.

Proposition 1.1.8. The product in the category of monoids is given by the product monoid; that is, if (M ) is a family of monoids such that for every i I there is a monoid homomorphism i i∈I ∈ 12 1 Basic concepts of binoids q : Q M , then there exists a unique monoid homomorphism q : Q M with π q = q for i → i → i∈I i k k all k I, where π denotes the canonical projection M M on the kth component. ∈ k i∈I i → k Q Proof. This is standard. Q

One can also consider the subset of the product i∈I Mi of a family (Mi)i∈I of binoids that contains all tuples (a ) with a =0 for almost all i I. Clearly, this is a subsemigroup of M . i i∈I i i ∈ Q i∈I i However, the next result gives reason for our convention to consider a binoid as a pointed set with Q distinguished element because, similar to rings, the direct sum of an infinite family (M , ) of ∞ i ∞i i∈I nonzero binoids as defined in Definition 1.1.6 is only a semibinoid.

Corollary 1.1.9. Let (Mi)i∈I be a family of binoids. With respect to the componentwise addition, the product M of the family (M , ) of pointed sets is a binoid with identity element (0 ) =: i∈I i i ∞i i∈I i i∈I 0 and absorbing element ( ) =: . The direct sum M is a semibinoid that coincides Π Q ∞i i∈I ∞Π i∈I i with the product if and only if M = for almost all i I. i {∞i} ∈ L Proof. This is similar to Lemma 1.1.7.

Lemma 1.1.10. The product i∈I Mi of a family (Mi)i∈I of nonzero binoids is commutative if and only if all M are commutative. i Q Proof. This is easily veriﬁed.

1.2 Homomorphisms

Deﬁnition 1.2.1. Let M and N be binoids (or semibinoids). A map ϕ : M N is a ( s e m i -) → b i n o i d h o m o m o r p h i s m if it is a monoid (resp. semigroup) homomorphism which sends to ∞M . Moreover, we call ϕ a monomorphism or embedding ifitsisinjective,an epimorphism ∞N if it is surjective, and an i s o m o r p h i s m if it is bijective. The set im ϕ := ϕ(M)isthe image of ϕ, and the set ker ϕ := a M ϕ(a)= { ∈ | ∞N } is the kernel of ϕ. The set of all binoid homomorphisms from M to N is denoted by hom(M,N).

Remark 1.2.2. A binoid homomorphism that satisﬁes ker = need not be injective. For example, {∞} the binoid homomorphism ϕ : N∞ 0, with x 0 if x = and otherwise, fulﬁlls ker ϕ = → { ∞} 7→ 6 ∞ ∞ , but is not injective. {∞} Example 1.2.3. Let M be a binoid. (1) There are canonical binoid homomorphisms

0, M . { ∞} −→ −→ {∞} (2) If M is commutative, the map M M with a ka is a binoid homomorphism for every k N. → → ∈ (3) Every element a M deﬁnes a binoid homomorphism ϕ : N∞ M with 1 a. Conversely, ∈ a → 7→ every binoid homomorphism N∞ M is determined by the image of 1. Hence, hom(N∞,M) = → ∼ M. The image im ϕ is given by the subbinoid ,na n N , and ker ϕ = if and only a {∞ | ∈ } a 6 {∞} if a is nilpotent. (4) If the operation on M is given by or , then M M c with A Ac is in either case a ∩ ∪ → 7→ binoid isomorphism. This holds particularly for the binoid (V ) and for the binoid deﬁned by S c a topology on V , for instance, for the discrete topology P(V ) ∼= P(V ) . 1.3 Generators 13

Example 1.2.4. Let I = 1,...,n , n 1. { } ≥ (1) The canonical bijections P 0, n, A (δ (A)) , andP 0, n, A (δ¯ (A)) , n,∩ →{ ∞} 7→ i i∈I n,∪ →{ ∞} 7→ i i∈I where 0 , if i A, , if i A, δi(A) = ∈ and δ¯i(A) = ∞ ∈ ( , otherwise, (0 , otherwise, ∞ for i I, are binoid isomorphisms. ∈ (2) Let (M ) be a family of binoids and k I. The projection on the kth component i i∈I ∈ M M , (a ) a , i −→ k i i∈I 7−→ k Yi∈I is a binoid epimorphism, whereas the inclusion M ֒ M given by k → i∈I i a ( ,..., , a, ,...,Q ) 7−→ ∞ ∞ ∞ ∞ is only a semibinoid homomorphism, and the inclusion given by

a (0,..., 0, a, 0,..., 0) 7−→ is only a monoid homomorphism (in either case a is the kth component). The category of binoids B possesses an initial object, namely the trivial binoid 0, . The terminal { ∞} object of B is given by the zero binoid . Note that there is no binoid homomorphism from the {∞} zero binoid into a nonzero binoid. The category of commutative binoids is a full subcategory of B which we denote by comB. Proposition 1.2.5. The product in the category of binoids B is given by the product. Proof. This follows immediately from Proposition 1.1.8 and Corollary 1.1.9.

The (ﬁnite) coproduct of commutative binoids will be described in Section 1.8. The following statement is a Cayley-type embedding theorem for binoids. Lemma 1.2.6. Given a binoid M, there exists a binoid embedding

M map M , x (t : y x + y) . −→ ∞ 7−→ x 7→ ′ Proof. This map is a binoid homomorphism because (t t ′ )(y) = x + x + y = t ′ (y) for all x ◦ x x+x x, y M, 0 t0 = idM , and (t∞ : y ). The injectivity follows from the simple fact that ∈ 7→ ∞ 7→′ 7→ ∞ ′ t = t ′ is equivalent to x + y = x + y for all y M, which implies x = x for y = 0. x x ∈

1.3 Generators

Now we consider generators of binoids and introduce free, free commutative, and semifree binoids as well as binoids defined by generators and relations. Later, cf. Section 1.12, finitely generated commutative binoids will be treated in more detail. Definition 1.3.1. Let M be a binoid and A M a subset. Since the intersection of a family of ⊂ subbinoids of M is a subbinoid, there exists a smallest subbinoid of M containing the set A which will be called the binoid g e n e r a t e d by A and denoted by

A . h i 14 1 Basic concepts of binoids

If M = A , we say M is generated by A and call A a generating set and its elements h i generators of M. In this case, A iscalleda minimal generating set of M if no proper subset of A generates M. A binoid is f i n i t e l y g e n e r a t e d if it is generated by a finite subset. A finitely generated binoid that admits a (minimal) generating subset with n elements and all other generating subsets consist of n elements is called n - g e n e r a t e d . A f i n i t e binoid consists of finitely many ≥ elements only. These definitions apply to a semibinoid S and a subset A S if they hold for the ⊆ binoid S0 and A.

A generating set A of a binoid M generates M as a monoid if and only if there are elements a,b M • ∈ with a + b = (this property will be called non-integral later). Otherwise, A generates M as a ∞ ∪{∞} monoid. In particular, a binoid is ﬁnitely generated if and only if it is ﬁnitely generated as a monoid.

Example 1.3.2. (1) The zero binoid is ﬁnitely generated by as a monoid and as a binoid. The trivial binoid {∞} ∅ 0, is generated as a binoid by , but its monoid generator is . { ∞} ∅ ∞ (2) N∞ = 1 as a binoid but the monoid generators of N∞ are 1 and . In general, (Nn)∞, n 1, h i ∞ ≥ is ﬁnitely generated as a binoid by the elements

ei := (0,..., 0, 1, 0,..., 0) ,

i 1,...,n , where 1 is the ith entry, and as a monoid by ( ,..., ) and e , i 1,...,n . ∈{ } ∞ ∞ i ∈{ } (3) If (M ) is a finite family of binoids and A M a generating set of M , i I, then M i i∈I i ⊆ i i ∈ i∈I i is generated by Q e := (0,..., 0, , 0,..., 0) and ae := (0,..., 0, a, 0,..., 0) , i,∞ ∞ i a A , i I, where and a are the ith component of e and ae , respectively. For instance, ∈ i ∈ ∞ i,∞ i (N∞)n, n 1, is generated as a monoid and as a binoid by the elements e and e , i I. ≥ i i,∞ ∈ (4) Subbinoids of a finitely generated binoid need not be finitely generated. For instance, the subbinoids ((N∞ )n)0 (N∞)n and ((N )n)0,∞ (Nn)∞ are not finitely generated for n 2 ≥1 ⊆ ≥1 ⊆ ≥ since every generating set has to contain the elements (n, 1,..., 1), n 1, for instance. ≥ (5) Let n,m N. The operation n m := lcm(n,m) on N yields a commutative binoid (N, , 1, 0) that ∈ ⋄ ⋄ is not finitely generated since every generating set has to contain all powers of prime numbers. (6) If a topology on V admits a basis B, then B generates the binoid ( , ,V, ) if and only if T T ∪ ∅ is a quasi-compact topology on V ; that is, when every open covering admits a finite one. For T instance, the binoid P(V )∩ defined by the discrete topology on V is finitely generated if and only if V is finite.

Lemma 1.3.3. A commutative binoid is finite if and only if it is finitely generated and every one- generated subbinoid is finite.

Proof. Only the if part of the statement is not trivial. Every finitely generated commutative binoid M = x ,...,x , r N, gives rise to a canonical binoid epimorphism h 1 ri ∈ r r x M with (n x ,...,n x ) n x . h ii −→ 1 1 r r 7−→ i i i=1 i=1 Y X The product is finite since all its components are finite by assumption. Now the finiteness of M follows from the surjectivity.

If M is a binoid generated by the (not necessarily ﬁnite) set A, every element f M • can be written ∈ 1.3 Generators 15 as a ﬁnite sum of the generators. In case M is commutative, we have the following notation

f = naa aX∈A with n N and n = 0 for almost all a A. Of course, this expression need not be unique. a ∈ a ∈ Deﬁnition 1.3.4. Let V be an arbitrary set of elements. Let M(V ) denote the free monoid consisting of all ﬁnite sums of elements in V with addition given by

(x + + x ) + (y + + y ) := x + + x + y + + y , 1 ··· n 1 ··· m 1 ··· n 1 ··· m x ,y V , i 1,...,n , j 1,...,m , and the sum over the empty set (:= 0) as identity element. i j ∈ ∈{ } ∈{ } The binoid M(V )∞ =: F(V ) is called the f r e e b i n o i d on V .

Lemma 1.3.5. Every element = of F(V ) can be written uniquely as a sum of elements of V . 6 ∞ Proof. This is clear since x + + x = (x )+ + (x ). 1 ··· n 1 ··· n Obviously, F(V ) is commutative if and only if V = x is a singleton. In a commutative binoid with { } more than one generator, one cannot expect unique expression of elements.

Lemma 1.3.6. Let M be a binoid. Every subset A = a i I M gives rise to a unique binoid { i | ∈ }⊆ homomorphism ε : F(I) M, i a , −→ 7−→ i i I, which is surjective if and only if A generates M. ∈ Proof. This is easily veriﬁed.

The preceding lemma implies, in particular, that every generating set A = a i I of a binoid M { i | ∈ } yields a canonical binoid epimorphism ε : F(I) M with i a , i I. → 7→ i ∈ Definition 1.3.7. Let M be a commutative binoid. We say M is a f r e e c o m m u t a t i v e binoid if there exists a binoid isomorphism ε : (N(I))∞ M for some (possibly infinite) set I. The family → (ε(ei))i∈I of elements in M is called a b a s i s of M. The free commutative binoid with basis V will be denoted by FC(V ) or by FC if V = 1,...,n . n { } Lemma 1.3.8. Every element f = of a finitely generated free commutative binoid with basis (x ) 6 ∞ i i∈I admits a unique expression as f = n x with n =0 for almost all i I. i∈I i i i ∈ Proof. This follows from the fact thatP every element a (N(I))∞ has this property for I finite. ∈ Example 1.3.9. The commutative binoid Z∞ is not free commutative because there are non-trivial invertible elements, which yield non-unique expressions of 0 Z∞. The canonical binoid epimorphism ∈ ϕ : (N2)∞ Z∞ with (1, 0) 1 and (0, 1) 1 is not injective because ϕ−1(0) = (n,n) n Z → 7→ 7→ − { | ∈ } for instance. See also Example 1.3.13(2).

Lemma 1.3.6 carries over to free commutative binoids.

Lemma 1.3.10. Given a family (ai)i∈I of commuting elements in a binoid M, i.e. ai + aj = aj + ai for all i, j I, there exists a unique binoid homomorphism ∈ ε : FC(I) M, i a , −→ 7−→ i i I, which is surjective if and only if a i I generates M. ∈ { i | ∈ } 16 1 Basic concepts of binoids

Proof. All statements are easily veriﬁed.

In a commutative binoid M every set a i I of elements in M gives rise to a binoid homomor- { i | ∈ } phism (N(I))∞ M, e a , −→ i 7−→ i i I, that is surjective if and only if a i I generates M. For instance, ∈ { i | ∈ } ϕ : (Nn)∞ (N∞)n , e e , −→ i 7−→ i i 1,...,n , is not surjective because e ,...,e is only half of a generating set of (N∞)n, cf. ∈ { } { 1 n} Example 1.3.2(3). Note that

π : (N∞)n (Nn)∞ with e e and e , −→ i 7−→ i i,∞ 7−→ ∞ i 1,...,n , is an epimorphism such that πϕ = id Nn ∞ . ∈{ } ( ) Definition 1.3.11. Let V be an arbitrary set. The binoids that arise from F(V ) and FC(V ) by taking additional relations , i I, among the elements of V into account will be denoted by Ri ∈ F(V )/( ) and FC(V )/( ) . Ri i∈I Ri i∈I Here we tacitly assume the reader to be familiar with the characterization of a monoid defined by generators and relations given in the above definition in a little sketchy way. A precise justification of the notation (in the commutative case) can be found in Example 1.6.7. Definition 1.3.12. Let M be a nonzero commutative binoid. We say M is a s e m i f r e e binoid with semibasis (a ) if M is generated by a i I such that every element f M • can be written i i∈I { i | ∈ } ∈ uniquely as f = i∈I niai with ni = 0 for almost all i I. Then the set ai ni =0 =: supp(f) is ∈ { | 6 0 } called the s u p p o r t of f. A commutative semibinoid S is semifree if the binoid S is semifree. P Obviously, every finitely generated free commutative binoid is semifree, cf. also Corollary 1.4.28, and a semibasis is always a minimal generating set. Semifree binoids represent an important class of commutative binoids, for this see the characterization in Corollary 2.1.21. Example 1.3.13. (1) The binoid (N(I))∞ is semifree with semibasis e , i I. i ∈ (2) The canonical generators 1 and 1 of Z∞ form no semibasis since 0 = n 1+ n ( 1) for all − · · − n 1. In fact, Z∞ is not semifree. Every minimal generating set of Z∞ is given by two integers ≥ n,m Z with n > 0, m < 0, and gcd(n, m) = 1. Hence, kn + lm = 1 for some k,l > 0, ∈ − which yields kn˜ + ˜lm = 1 for some k,˜ ˜l > 0 by adding mn nm = 0 sufficiently often to − − kn lm = 1. By applying this to the equations above, we obtain a non-unique expression of − − − 0 in terms of n and m. See also Lemma 1.4.23. (3) The semibasis of the free commutative binoid (N, , 1, 0) is given by p p N prime by the · { | ∈ } fundamental theorem of arithmetic, cf. [51, Hauptsatz 10.1]. (4) A commutative binoid with a non-trivial idempotent (see below, Section 1.4) is never semifree. For instance, P(V ) is not semifree since A = A A = A A A = for every A V . ∪ ∪ ∪ ∪ ··· ⊆ However, the singletons v , v V , V finite, generate P(V ) and every set A V is uniquely { } ∈ ∪ ⊆ given as A = v . Thus, P(V ) can be considered as semifree up to idempotence. In Section ∪v∈A{ } ∪ 6.1, we will encounter these kind of binoids. (5) The generating set A = 1/pn1 pnr p ,...,p N prime, n ,...,n 1, r N of { 1 ··· r | 1 r ∈ 1 r ≥ ∈ } (Q∞ , +, 0, ) is no semibasis because, for instance, 1 = 1 + 1 = 1 + 1 + 1 + 1 = .... Since ≥0 ∞ 2 4 4 8 8 8 8 this sequence is infinite, one cannot deduce a semibasis from A by omitting elements. 1.4 Additivity properties 17

(6) The binoid FC(x , x )/(x + x = ) is semifree because every element = can be written 1 2 1 2 ∞ 6 ∞ uniquely as nx or mx for certain n,m 0. This example shows that Lemma 1.3.10 need not 1 2 ≥ be true for semifree binoids since there exists, for instance, no binoid homomorphism

FC(x , x )/(x + x = ) FC(y ,y ) 1 2 1 2 ∞ −→ 1 2 with x y , i 1, 2 . i 7→ i ∈{ } (7) Another interesting class of semifree binoids are simplicial binoids, which will be introduced in Section 6.5.

1.4 Additivity properties

This section deals with additivity properties of binoids. Before we recall classical concepts that originate from monoid theory, we point out those properties of a binoid which have no counterpart when there is no absorbing element, namely non-integrity and nilpotence. Concerning those properties that translate directly from monoids to binoids we follow essentially [28]. Then the additively closed subobjects of a binoid, its ﬁlters, are introduced. These are very useful when we are missing the theory of (prime) ideals of a binoid. The connection will be shown in Chapter 2. We close this section with the behavior of additive properties under homomorphisms. The phenomena that may occur if an absorbing element exists are described in the following deﬁni- tions.

Deﬁnition 1.4.1. Let M be a binoid (or semibinoid). An element a M is called n i l p o t e n t if ∈ na = a + + a = for some n 1. The set of all nilpotent elements will be denoted by nil(M). We ··· ∞ ≥ say M is r e d u c e d if nil(M)= , and M is s t r o n g l y r e d u c e d if a + a + b = for a,b M {∞} ∞ ∈ implies a + b = . ∞ Lemma 1.4.2. (1) A strongly reduced binoid is reduced. (2) A commutative binoid is strongly reduced if and only if it is reduced.

Proof. (1) If na = for some a and n 2 in a strongly reduced binoid, then by applying successively ∞ ≥ = na = a + a + (n 2)a = a + (n 2)a = (n 1)a , ∞ − − − we obtain = 2a = a + a + 0, which yields = a +0= a. (2) Any equation a + a + b = in a ∞ ∞ ∞ commtative binoid yields 2(a + b)= by adding b. Hence, a + b = if the binoid is reduced. ∞ ∞ Deﬁnition 1.4.3. Let M = be a binoid (or semibinoid). An integral element a M • satisﬁes 6 {∞} ∈ the property that a + b = or b + a = implies b = . The set of all integral elements will be ∞ ∞ ∞ • denoted by int(M) and the complement M int(M) by intc(M). We say M is i n t e g r a l if M is a \ • submonoid (resp. subsemigroup) of M; that is, M consists only of integral elements.

Every binoid that emerges from a monoid by adjoining an absorbing element is integral, and all integral binoids are of this type.

Example 1.4.4. (1) By deﬁnition, (Nn)∞ is an integral binoid and (Zn)∞ is a binoid group (see below). On the other hand, the binoids (N∞)n and (Z∞)n, n 2, are obviously not integral. ≥ 18 1 Basic concepts of binoids

(2) The notion of integrity for a ring R and for its underlying binoid (R, , 1, 0) coincides; that is, · the non-integral elements are precisely the zero-divisors of R. Therefore, intc(R) = 0 if and { } only if R is an (integral) domain. (3) A binoid defined by a topology on V is integral if and only if V is irreducible with respect to T∩ ; that is, V = and U V = for arbitrary nonempty open subsets U, V . T 6 ∅ ∩ 6 ∅ ⊆ T For instance, the binoid defined by the Zariski topology on An and An(K) for n 1 and K an K ≥ algebraically closed field, is integral as well as the binoids given by the subspace topology on an (affine) variety V An , cf. [32, Chapter I.1]. ⊆ K (4) The binoid P(V ) is never integral for #V 2 since A Ac = for every subset A V . ∩ ≥ ∩ ∅ ⊂ Similarly, P(V ) with #V 2 is never integral. From the criterion given in (3) we can easily ∪ ≥ derive non-trivial integral subbinoids of P(V ) for #V 2, like Y, x ,..., Y, x P(V ) , ∩ ≥ h{ 1} { k}i ⊆ ∩ where = Y ( V such that V Y = x ,...,x with k 1. ∅ 6 \ { 1 k} ≥ (5) The set of integral and non-integral elements in the product of a family (Mi)i∈I of binoids are given by

int Mi = int(Mi) Yi∈I Yi∈I and intc M = (a ) a intc(M ) for at least one k I . i { i i∈I | k ∈ k ∈ } Yi∈I Lemma 1.4.5. Nilpotent elements are not integral; that is, nil(M) intc(M) for every nonzero ⊆ binoid M. In particular, an integral binoid is reduced.

Proof. This follows directly from the deﬁnitions.

Lemma 1.4.6. The subset M := int(M) M is an integral subbinoid for every nonzero int ∪{∞} ⊆ binoid M.

Proof. This is easily veriﬁed.

Example 1.4.7. The binoid M := FC(x, y)/(x + y = ) has no non-trivial integral elements (i.e. ∞ M = 0, ), but two non-trivial integral subbinoids, namely x and y . int { ∞} h i h i Now we translate common properties of monoids (resp. semigroups) and their elements to binoids. We start with the notions of units and idempotents which are independent of the existence of an absorbing element.

Deﬁnition 1.4.8. Let M be a monoid (or nonzero binoid). An element u in M is a u n i t if there exists an element a M such that a + u = u + a = 0. The element a is the unique (additive) i n v e r s e ∈ of u and will be denoted by -u. The set of all units M × is a submonoid of M which is a group, the × unit group of M. The set of all nonunits M M will be denoted by M+. We say M is positive if \ • × it has a trivial unit group (i.e. M 0 = M+). A b i n o i d g r o u p is a binoid G such that G = G ; • \{ } that is, G is a group.

Among many other names for the property of a monoid or binoid to have a trivial unit group pointed in combinatorics ([41]) and sharp in geometry ([45]) are also very common, while we follow [11]. We ﬁx the following notation: if R is a ring, then R∞ denotes the (commutative) binoid group that arises by adjoining an absorbing element to R with respect to its additive structure. For instance,

(Zn)∞ = (Zn , +, (0,..., 0), ) and (Z/mZ)∞ = (Z/mZ , +, [0], ) , ∪ {∞} ∞ ∪ {∞} ∞ where n 1, and m 2. ≥ ≥ 1.4 Additivity properties 19

× × Example 1.4.9. If (Mi)i∈I is a family of nonzero binoids, then i∈I Mi = i∈I Mi . Lemma 1.4.10. Every unit is integral; that is, M × int(M) forQ a nonzero binoidQ M. ⊆ Proof. Obvious.

Deﬁnition 1.4.11. Let M be a semigroup. An element f M is called i d e m p o t e n t if f + f = f. ∈ In a b o o l e a n semigroup every element is idempotent. The set of all idempotent elements will be denoted by bool(M). A commutative semigroup that is boolean is called a s e m i l a t t i c e .

The operation of a semilattice L is usually denoted by or and L is called a j o i n - or m e e t - ⊔ ⊓ semilattice, respectively. In either case, the operation gives rise to a partial order on L by setting ≤ x y : x y = y or x y : x y = x. ≤ ⇔ ⊔ ≤ ⇔ ⊓ Example 1.4.12. (1) The identity element and the absorbing element are always idempotent elements. In particular, the set of idempotents of a commutative binoid M is a subbinoid since 2(a + b)=2a +2b = a + b for idempotent elements a,b M. More precisely, bool(M) is the largest boolean subbinoid of ∈ a commutative binoid M. (2) The set P(V ) is a meet- and join-semilattice with respect to and , respectively. ∩ ∪ (3) The commutative binoid deﬁned by a topology is boolean such that ( ) is the induced partial ⊆ ⊇ order with respect to the operation ( ). Moreover, Example 1.4.4(3) and (4) show that a ∩ ∪ boolean binoid can be integral in contrast to a boolean algebra = 0 , which has characteristic 6 { } 2 always.

Lemma 1.4.13. Every boolean binoid is positive and reduced.

Proof. Only the positivity of a boolean binoid need to be shown. For this let a + b = 0 for two idempotent elements a and b. Adding a from the left and b from the right yields a = a + b = b, hence a = b = 0.

The well-known notion of torsion and cancellativity in monoid theory need to be modiﬁed for binoids.

Deﬁnition 1.4.14. Let M be a binoid (or semibinoid). An element a in M is a t o r s i o n element if a = or na = nb for some b M with b = a and n 2. We say M is torsion-free if there areno ∞ ∈ 6 ≥ other torsion elements in M besides ; that is, na = nb implies a = b for every a,b M and n 1. ∞ ∈ ≥ If na = nb = implies a = b for every a,b M and n 1, then M is called t o r s i o n - f r e e u p t o 6 ∞ ∈ ≥ nilpotence.

By deﬁnition, a binoid is torsion-free if and only if it is reduced and torsion-free up to nilpotence. With this notation, a monoid with no absorbing element is torsion-free if M ∞ is a torsion-free binoid. A group G is a torsion group if and only if all elements of G∞ are torsion elements. In particular, the unit group M × need not be torsion-free. An important example of binoids that are torsion-free up to nilpotence but (in general) not reduced is given in Corollary 2.1.21. The set of all torsion elements in a binoid that is torsion-free up to nilpotence is precisely nil M. In general, the set of all torsion elements in M has no structure as the following example shows.

Example 1.4.15. Consider the binoid M = FC(x, y)/(10x +2y = ). The elements x and y are no ∞ torsion elements but every element nx + my with n,m 1 is a torsion element. ≥ Lemma 1.4.16. (1) Nilpotent elements are torsion elements. (2) Boolean binoids are torsion-free. 20 1 Basic concepts of binoids

Proof. Both statements follow directly from the deﬁnitions.

The second statement of the preceding lemma shows that there are non-trivial ﬁnite binoids which are torsion-free.

Deﬁnition 1.4.17. Let M be a commutative monoid (or binoid). Two elements a,b M are called ∈ asymptotically equivalent if there exists an n N with na = nb for all n n . We say 0 ∈ ≥ 0 that M is free of asymptotic torsion if any two distinct elements of M are not asymptotic equivalent.

Of course, asymptotically equivalent elements are torsion elements and nilpotent elements are asymptotically equivalent to . ∞ Lemma 1.4.18. Let M be a commutative binoid and a,b M. The following conditions on a and b ∈ are equivalent: (1) a and b are asymptotically equivalent. (2) na = nb and (n + 1)a = (n + 1)b for an n N. ∈ (3) na = nb and ma = mb for some n,m N with gcd(n,m)=1. ∈ Proof. The implications (1) (2) (3) are trivial. For (3) (1) consider the set N := l N∞ ⇒ ⇒ ⇒ { ∈ | la = lb . Obviously, N N∞ is a subbinoid because all possible combinations of elements in N lie } ⊆ in N. Now take n,m N with gcd(n,m) = 1. By [28, Theorem 2.2], there are for every k N with ∈ ∈ k (n 1)(m 1) non-negative integers r and s such that k = rn + sm N. Hence, ka = kb for all ≥ − − ∈ k (n 1)(m 1). ≥ − − A weaker property than being asymptotically equivalent, called functionally equivalent, will come into play at the end of Section 4.4.

Deﬁnition 1.4.19. Let M = be a binoid (or semibinoid). An element a M • is called 6 {∞} ∈ cancellative if from b + a = c + a = or a + b = a + c = , for b,c M it follows b = c. We say 6 ∞ 6 ∞ ∈ M is c a n c e l l a t i v e if every element = is cancellative. The set of all cancellative elements in M 6 ∞ will be denoted by can(M). M is called r e g u l a r if it is integral and cancellative element.

A cancellative binoid has no non-trivial idempotent elements. The trivial binoid is regular, whereas the zero binoid was excluded from the deﬁnitions of an integral and cancellative binoid. Note that a cancellative element in a binoid need not be integral, it may even be nilpotent. For instance, the binoid FC(x)/(nx = ) ∞ with n 2 is cancellative. Thus, for binoids, cancellativity is far from being such a strong property ≥ as it is for monoids. The equivalent property for binoids is regularity because a binoid M is regular if and only if M • is a cancellative monoid; that is, M • is a monoid in which b+a = c+a or a+b = a+c implies b = c for all a,b,c M. See also Lemma 1.4.21(2) and Proposition 1.7.14 with the subsequent ∈ Remark 1.7.15. Similar to a cancellative commutative monoid, which can always be embedded in a commutative group, cf. [28, Theorem 1.2], any regular commutative binoid M admits an embedding .M ֒ G into a commutative binoid group G. For this result see Proposition 1.13.11 → Example 1.4.20. The binoid P(V ) is cancellative for #V < 3 and not cancellative for #V 3 ∩ ≥ since 1, 2 1, 3 = 1, 2 1 . By a similar argument, a subbinoid N P(V ) is not cancellative { }∩{ } { }∩{ } ⊆ ∩ if there are A, B N with A B = . Analogous statements hold for P(V ) . ∈ ∩ 6 ∅ ∪ Lemma 1.4.21. Let M be a binoid. (1) can(M)∞ is a cancellative subbinoid of M. 1.4 Additivity properties 21

(2) Units are cancellative. Conversely, if M is regular and ﬁnite, then M is a binoid group.

Proof. Both statements are easily veriﬁed except the converse of (2), which follows from the same argument that proves the analogous statement in group theory.

Example 1.4.22. The binoid FC(x, y)/(x + y = y, 2x = x) has no non-trivial cancellative elements (i.e. can(M)∞ = 0, ), but a non-trivial cancellative subbinoid, namely y . { ∞} h i Lemma 1.4.23. Semifree binoids are positive, cancellative, and torsion-free up to nilpotence.

Proof. This is easily veriﬁed.

The converse of the preceding lemma is false. For instance, the binoid

M := FC(x, y)/(2x =3y) is positive, torsion-free, and cancellative but not semifree. Note that x, y is a minimal generating { } • set of M (which is unique by Proposition 1.4.27 below) and that every element f M can be written ∈ uniquely as f = my or f = x + my for some m N. The positivity is clear. To show that M is torsion-free, one only needs to check ∈ that an equation like k(x + my) = kny for some k 2 is not possible. This is true since otherwise ≥ it would imply k = 2k′ and therefore yield the impossible equation 3 + 2m = 2n. Similarly, one can show that the generators x and y are cancellative elements, which implies the cancellativity of M by the following lemma.

Lemma 1.4.24. A binoid is integral or cancellative if and only if a generating set is so.

Proof. The only if part of the assertion is obvious. So let A M be a generating set consisting of ⊆ integral elements and suppose that a+b = for some a,b M. Let b = so that b = ˜b+x for some • ∞ ∈ 6 ∞ ˜b M and x A. Hence, = a + b = a + ˜b + x, which implies that a + ˜b = by the integrality ∈ ∈ ∞ ∞ of x. Proceeding the same way yields a + y = for some y A, hence a = . The assertion for the ∞ ∈ ∞ cancellativity of M follows similarly.

Even if the generators of a binoid are no torsion elements, the binoid may not be torsion-free. Consider, for instance, the binoid FC(a,b,c,d)/(n(a + b) = n(c + d)) with n 2. Also, a binoid ≥ may have non-trivial cancellative elements though all generators are not cancellative as the following example shows.

Example 1.4.25. The binoid M := FC(x, y)/(3x = 3y = , 2x + y = x + 2y) is positive, not ∞ reduced (hence not integral), and not cancellative since both generators x and y are not cancellative for instance. However, a := 2x + y is a non-trivial cancellative element because

(2x + y)+ x = and (2x + y)+ y = (x +2y)+ y = x +3y = , ∞ ∞ and so there are no equations like a + b = a + c = with b,c M. 6 ∞ ∈ Lemma 1.4.26. If M is an integral commutative binoid that admits a generating set of non-cancellative elements, then M contains no non-trivial cancellative element.

Proof. Let A can M be a generating set of M. By the assumption on A, there is for every f M • ⊆ ∈ an a A with f = f˜+ a and a + b = a + c = for some c,b M with c = b. Since M is integral the ∈ 6 ∞ ∈ 6 latter equation implies that f˜+ a + b = f˜+ a + c = by adding f˜. Hence, f + b = f + c = with 6 ∞ 6 ∞ c = b. 6 22 1 Basic concepts of binoids

The preceding result may fail to be true for an integral binoid that is not commutative, like the binoid F(x,y,z)/(z + x = z + y = x + y = x + z) in which the element x + y is cancellative.

Proposition 1.4.27. Every ﬁnitely generated commutative binoid M that is positive and cancellative admits a unique minimal generating set given by M 2M . + \ + Proof. By the positivity, every generating set is contained in M = M 0 . First we show that the + \{ } set M 2M contains any minimal generating set, which implies that it is a generating set as well. + \ + For this take an arbitrary minimal generating set x ,...,x and assume that x M 2M . Then { 1 r} 1 6∈ + \ + x = y + z for some y,z M . Hence, x = n x + + n x with at least two n 1 or one n 2, 1 ∈ + 1 1 1 ··· r r i ≥ i ≥ i I. If n = 0, the cancellativity of M yields a non-trivial equation 0 = (n 1)x +n x + +n x ∈ 1 6 1 − 1 2 2 ··· r r which is a contradiction to M × = 0 . Consequently, n = 0 and x can be dropped, contrary to the { } 1 1 minimality of the generating set. Thus, x ,...,x M 2M . { 1 r}⊆ + \ + The minimality of M 2M follows immediately because if x M 2M could be omitted, there + \ + ∈ + \ + were an expression x = n y + + n y with y M 2M and at least one n = 0, i 1,...,s . 1 1 ··· s s i ∈ + \ + i 6 ∈{ } This means x = y for some i 1,...,s since x 2M , hence x cannot be omitted. The same i ∈ { } 6∈ + argument shows that M 2M must be contained in every generating set of M. + \ + All conditions on a binoid assumed in Proposition 1.4.27 are required. Counterexamples of binoids that fulﬁll all except one of the three conditions are given by (Q∞ , +), (Z/nZ)∞ with n 2, and ≥0 ≥ FC(x, y)/(x + y = x).

Corollary 1.4.28. A ﬁnitely generated binoid is free commutative if and only if it is commutative, integral and semifree.

Proof. All properties of a free commutative binoid follow from the fact that (Nn)∞ has these properties. For the converse note that a binoid M with the given properties admits a unique minimal generating set by Proposition 1.4.27, say M 2M = x ,...,x . Since M is semifree and integral, the + \ + { 1 n} canonical binoid epimorphism (Nn)∞ M with e x , i 1,...,n , is injective. → i 7→ i ∈{ } The proof of Proposition 1.4.27 shows that the set M 2M is finite. This can be generalized. + \ + Lemma 1.4.29. Let M be a positive finitely generated commutative binoid. Then M nM is a + \ + finite set for all n 1. ≥ Proof. If x ,...,x is a minimal generating set of M, then x ,...,x M by the positivity of M, { 1 r} 1 r ∈ + and hence nM = n x + + n x r n n . This gives + { 1 1 ··· r r | k=1 i ≥ } P n−1 k #(M+ nM+) # (n1,...,nr) n1 + + nr

H( ,M): N N − −→ with H(n,M) := #(M nM+) for n 1 and H(0,M) := 0, is called the Hilbert-Samuel function \ ≥n of M and H(M) := n∈N H(n,M)T the Hilbert-Samuel series of M. H(1,M)= 1 alwaysP holds. − Remark 1.4.31. As we will see later in Example 2.1.18(2), the deﬁnition of the Hilbert-Samuel function is similar to that of a semilocal ring, cf. [53, Chapter II.B 4]. § 1.4 Additivity properties 23

Definition 1.4.32. Let P(V ) be a collection of subsets of an arbitrary set V . The collection A⊆ A is called s u b s e t - c l o s e d (resp. superset-closed) if B A (resp. A B) for some B P(V ) and ⊆ ⊆ ∈ A , implies B ; that is, with every set all its subsets (resp. supersets) lie in . The set of all ∈ A ∈ A A subset-closed subsets of P(V ) will be denoted by (V ). S Lemma 1.4.33. Let V = be an arbitrary set. 6 ∅ (1) (V ) defines a topology on P(V ). In particular, (V ) defines two boolean commutative binoids S S with respect to and , namely ∪ ∩ ( (V ), , , P(V )) =: (V ) and ( (V ), , P(V ), ) =: (V ) , S ∪ {∅} S ∪ S ∩ {∅} S ∩ which are integral if and only if #V =1. (2) A collection P(V ) is subset-closed if and only if P(V ) is superset-closed. In particular, A⊆ \ A the set c(V ) := P(V ) (V ) of all superset-closed subsets of P(V ) defines two S { \A|A∈S } boolean commutative binoids with respect to and , namely ∪ ∩ ( c(V ), ,V, P(V )) =: c(V ) and ( c(V ), , P(V ), V ) =: c(V ) . S ∪ S ∪ S ∩ S ∩ Proof. The first statement of (1) is easily verified. For the supplement note that if there are i, j V ∈ with i = j, then , i , j = and P(V i ) P(V j ) = P(V ). (2) Note that if 6 {∅ { }}∩ {∅ { }} {∅} \{ } ∪ \{ } (V ) is subset-closed, then A P(V ) means A . Hence, B for all B P(V ) with A ∈ S ∈ \ A 6∈ A 6∈ A ∈ A B if is subset-closed and A . This is equivalent to B P(V ) for all B P(V ) with ⊆ A 6∈ A ∈ \ A ∈ A B. Hence, P(V ) is superset-closed. The supplement of (2) follows from (1). ⊆ \ A The cardinality of the set of all subset-closed sets significantly increases as V becomes larger. A detailed computation shows, for instance, # ( 1 ) = 2, # ( 1, 2 ) = 4, # ( 1, 2, 3 ) = 19, and S { } S { } S { } # ( 1, 2, 3, 4 )= 170 S { } Now we introduce the common concept of filters to the theory binoids. Definition 1.4.34. A subset F of a binoid M is a f i l t e r if the following conditions are satisfied. (1) 0 F . ∈ (2) If f,g F , then f + g F . ∈ ∈ (3) If f + g F , then f,g F . ∈ ∈ The filtrum of M is the set of all filters in M and will be denoted by (M). F Loosely speaking, F M is a filter if it is a submonoid with the property f + g F f,g F . ⊆ ∈ ⇒ ∈ The additional condition on a submonoid to be a filter may be called summand-closed. Since 0 is a summand of every element of M, the first condition can be replaced by F = . 6 ∅ Remark 1.4.35. If R is a ring, then every proper filter of (R, , 1, 0) is the complement of an arbitrary · union of (ring) prime ideals of R, and conversely, every complement of a filter is the union of (ring) prime ideals. Of course, this union does not need to be unique. For instance, the set of invertible elements R× is the complement of the union of all prime ideals as well as the complement of the union of all maximal ideals. After establishing the necessary ideal theory for binoids in Chapter 2, we will show that similarly, every filter = M in a binoid M is the complement of one unique prime ideal, cf. 6 Corollary 2.2.6. Therefore, considering proper filters or prime ideals in binoids amounts to the same theory.

If (Fi)i∈I is a family of filters in M, then i∈I Fi is again a filter in M, and since every unit is a summand of 0 the filter M × is contained in any other filter of M. Hence, the commutative binoid T (M) := ( (M), ,M,M ×) F ∩ F ∩ 24 1 Basic concepts of binoids is a (meet-) semilattice with largest element M and smallest element M × with respect to set inclusion. We will encounter this binoid frequently.

Deﬁnition 1.4.36. Let A be a subset of M. The ﬁlter Filt(A) generated by A is the set of all summands of sums that lie in A; in other words,

Filt(A) = F. A⊆F\∈F(M) By definition, it is the smallest filter of M containing A. The filter generated by a singleton A = f , { } f M, will be denoted by Filt(f). ∈ Remark 1.4.37. Since M is the only filter containing , we have Filt(f) = M if and only if f is ∞ nilpotent. Moreover, g Filt(f) means g = n f + + n f for some summands f ,...,f of f. ∈ 1 1 ··· r r 1 r Therefore, g Filt(f) is equivalent to g + x = nf for some suitable x M and n N. ∈ ∈ ∈ The operation F ⋆G := Filt(F +G), F, G (M), on (M) turns the set (M) into a commutative ∈F F F binoid, namely (M) := ( (M),⋆,M ×,M) . F ⋆ F This binoid will appear later in Proposition 1.13.9.

Example 1.4.38. Let V be a finite set. A subset F P(V ) is a filter in P(V ) if and only if F is ⊆ ∩ superset-closed (which implies that V F ) and contains only one minimal element J P(V ) with ∈ ∈ respect to . Thus, for every filter F , one has F = Filt(J) = A P(V ) J A for some J V . ⊆ { ∈ | ⊆ } ⊆ Similarly, a subset F P(V ) is a filter in P(V ) if and only if F is subset-closed (which implies that ⊆ ∪ F ) and contains only one maximal element J P(V ) with respect to . Thus, for every filter F , ∅ ∈ ∈ ⊆ one has F = Filt(J) = A P(V ) A J = P(J) for some J V . In particular, the one-to-one { ∈ | ⊆ } ⊆ correspondences

(P(V ) ) P(V ) (P(V ) ) F ∩ ←→ ←→ F ∪ F min F 7−→ ⊆ A P(V ) J A [ J A P(V ) A J = P(J) { ∈ | ⊆ } ←− 7−→ { ∈ | ⊆ } max F [ F ⊆ ←− yield the binoid isomorphisms P(V ) = (P(V ) ) and P(V ) = (P(V ) ) . ∪ ∼ F ∩ ∩ ∩ ∼ F ∪ ∩ Finally we take a look on the behavior of certain properties under a binoid homomorphism.

Lemma 1.4.39. Let ϕ : N M be a binoid homomorphism. → (1) ϕ(nil(N)) nil(M). ⊆ (2) ϕ(bool(N)) bool(M). ⊆ (3) ϕ(N ×) M ×. Moreover, if M is commutative, then ϕ−1(M ×) (N). ⊆ ∈F Proof. We only show the second part of the last assertion because all other statements are easily veriﬁed. Set F := ϕ−1(M ×). Clearly, 0 F because ϕ is a binoid homomorphism. If a,b F , N ∈ ∈ then ϕ(a), ϕ(b) M ×. Since M × is a group, we obtain ϕ(a + b)= ϕ(a)+ ϕ(b) M ×, which means ∈ ∈ a + b F . Similarly, a + b F means ϕ(a + b) M ×. So there is an element u M × with ϕ(a + b)= ∈ ∈ ∈ ∈ ϕ(a)+ ϕ(b) = u. This yields ϕ(a) + (ϕ(b)+(-u)) = 0M = ϕ(b) + (ϕ(a)+(-u)), which shows that ϕ(a), ϕ(b) M ×. ∈ 1.5 N - spectra 25

Remark 1.4.40. Note that the image of a non-integral element need not be non-integral under a binoid homomorphism. If, for instance, a N nil(N) is a non-integral element such that the set ∈ \ A := b N a + b = intc(N) is contained in the kernel of ϕ : N M, then ϕ(a) may even be { ∈ | ∞} ⊆ → a unit, cf. Example 1.5.5(2). A trivial example is given by the binoid homomorphism

FC(x, y)/(x + y = ) (Z/nZ)∞ , ∞ −→ n 2, with x 1 and y . ≥ 7→ 7→ ∞ Proposition 1.4.41. Let M be a binoid. A map ϕ : M 0, is a binoid homomorphism if and →{ ∞} only if M ker ϕ (M). \ ∈F Proof. Let F := M ker ϕ. We have ϕ(a + b) = 0 if and only if ϕ(a)=0= ϕ(b) since ϕ is a binoid \ homomorphism and, in particular, ϕ(0 )=0 . Hence, 0 F , and a + b F is equivalent to M N M ∈ ∈ a,b F . ∈ In Section 2.2, we will rephrase this result in terms of ideal theory, cf. Lemma 2.2.5.

1.5 N - spectra

The notion of homomorphisms between binoids yields the dual and bidual of a binoid as well as its N - spectrum for another binoid N. The latter admits a geometric interpretation when N = K is a field, namely the K - spectrum of the associated binoid algebra, which will be treated in Section 4.4 in more detail. All these objects are illustrated by several examples. By definition, every binoid homomorphism ϕ between nonzero binoids fulfills 0, im ϕ. Those { ∞} ⊆ homomorphisms ϕ with im ϕ = 0, are uniquely determined by ker ϕ and closely related to the { ∞} prime ideals of the binoid, cf. Section 2.2.

Deﬁnition 1.5.1. Let M be a binoid and N an arbitrary subset of M. The map χ : M 0, N →{ ∞} with a 0 if a N and otherwise, is called the i n d i c a t o r f u n c t i o n of N. The anti- 7→ ∈ ∞ indicator function χM\N of N will be denoted by αN . If an (anti-) indicator function is a binoid homomorphism we call it an (anti-) indicator homomorphism.

The indicator function is also known as the characteristic function and the codomain is usually the trivial binoid with respect to the multiplication; that is, χ , N M, is usually deﬁned to be the map N ⊆ M 1, 0 with a 1 if a N and 0 otherwise. →{ } 7→ ∈ The set map(M, 0, ) can be naturally embedded into the set map(M,M ′) for any two nonzero { ∞} binoids M and M ′; that is, every function M 0, can be considered as a map M M ′. Then →{ ∞} → we will use the same notation χ and α for the indicator and anti-indicator function of N M in N N ⊆ map(M,M ′), respectively.

Remark 1.5.2. Let M and N be nonzero binoids. The set hom(M,N) is a semigroup with respect to the operation (φ + ψ)(a) = φ(a)+ ψ(a) , φ, ψ hom(M,N), a M. The semigroup hom(M,N) is commutative if M = 0, or if N ∈ ∈ { ∞} is commutative. If N is boolean, then so is hom(M,N). In particular, the set of all indicator homomorphisms on M is a boolean subsemigroup of hom(M,N). We have χ • + ψ = ψ + χ • = ψ and χ + ψ = ψ + χ = χ for all ψ : M N. In M M {0} {0} {0} → general, the functions χM • ,χ{0} : M N are no homomorphisms. More precisely, χM • is a binoid • → homomorphism if and only if M is a monoid, and χ{0} is a binoid homomorphism if and only if M is 26 1 Basic concepts of binoids

• positive. Thus, (hom(M,N), +,χM • ,χ{0}) is a binoid if and only M is a positive monoid. However, one can always adjoin an absorbing and an identity element to get a binoid. For instance, the constant maps α : a and χ : a 0, a M, can serve as an absorbing and as an identity element, M 7→ ∞ M 7→ ∈ respectively.

Deﬁnition 1.5.3. Let M and N be binoids. The semigroup (hom(M,N), +) =: N-spec M is called the N - spectrum of M and we refer to an element of N-spec M as an N - point of M. With this notation, the semilattice of all indicator homomorphisms on M, which is contained in N-spec M, is denoted by 0, -spec M. By Proposition 1.4.41, { ∞} 0, -spec M = χ F (M) . { ∞} { F | ∈F }

The N - point χ × : M N is called the s p e c i a l p o i n t of M. In general, we refer to an N - point M → χ : M N given by a ﬁlter F (M) asa characteristic (or boolean) point of M. We F → ∈F use this terminology, in particular, if N is the trivial binoid.

Remark 1.5.4. The terminology introduced here becomes clear later when we will see that the filters in M are precisely the complements of the prime ideals in M, cf. Corollary 2.2.6. Hence, the special × point χ × = α corresponds to the maximal ideal M (maximal with respect to since M is the M M+ + ⊆ smallest filter and contained in any other filter), and the characteristic points χ = α , F (M), F M\F ∈F correspond to the the prime ideals M F . \ For a field K, the K - spectrum of a binoid M can be identified with the K - spectrum K-spec K[M] := HomK- alg(K[M],K) of K[M], cf. Propostition 3.2.3, which is a well-studied topological space if M is commutative. Section 4.4 and in most parts Chapter 5 are devoted to a detailed description of the set of K - points of M by applying the knowledge of K-spec K[M], where K is a field and M a commutative binoid. Once in a while, we anticipate this description of the K - spectrum of a binoid and draw a picture of the R - spectrum.

Example 1.5.5. Let N be a commutative binoid.

(1) By Lemma 1.4.39(2), N-spec FC(x)/(2x = x) ∼= bool N. (2) Every binoid homomorphism 0, n N is uniquely determined by the images of the gen- { ∞} → erators e1,∞,...,en,∞. Since the elements ei,∞ are idempotent their images lie in bool N, and because there are no further relations than e + + e = , we get 1,∞ ··· n,∞ ∞Π N-spec 0, n = (a ,...,a ) (bool N)n a + + a = . { ∞} { 1 n ∈ | 1 ··· n ∞} In particular, N-spec 0, = and 0, -spec 0, n = 0, n 0 , which one may { ∞} ∼ {∞} { ∞} { ∞} ∼ { ∞} \{ Π} also obtain from Proposition 1.8.14.

We have to postpone the determination of N-spec i∈I Mi for an arbitrary family (Mi)i∈I of binoids because a little more theory is needed for this, cf. Proposition 1.8.14. Q Proposition 1.5.6. Let M, N, and L be binoids. Every binoid homomorphism ϕ : M N induces → a semigroup homomorphism ϕ∗ : L- spec N L- spec M with ψ ψϕ. Moreover, ϕ∗ is injective if −→ 7→ ϕ is surjective.

Proof. The ﬁrst assertion is trivial. For the supplement, we need to show that ψ = ψ′ if ψϕ = ψ′ϕ, but this follows from the surjectivity of ϕ.

(I) ∞ (I) Lemma 1.5.7. If N is a nonzero binoid, then N- spec(N ) ∼= N as binoids. In particular, for every ﬁnitely generated free commutative binoid M there is a unique n 1 such that N- spec M = N n. ≥ ∼ 1.5 N - spectra 27

Proof. Since (N(I))∞ is integral, positive, and bool((N(I))∞) = 0, , the images of the elements {(I) ∞}∞ of the basis (ei)i∈I , which determine every homomorphism on (N ) , can be choosen at will in N. (I) ∞ Moreover, (N-spec(N ) , +,χM • , χ{0}) is a binoid by Remark 1.5.2. The supplement is clear by the deﬁnition of a free commutative binoid.

Lemma 1.5.8. Let M and N be binoids. If N is positive, the homomorphism χ × : M N is an M → absorbing element in N- spec M. In particular, N- spec M is a boolean semibinoid if N is a boolean binoid.

Proof. By Proposition 1.4.41, the characteristic function χM × is a binoid homomorphism. Since N is positive, ϕ(f) = 0 for all f M × and all ϕ N-spec M. It follows ∈ ∈ 0 + 0 , if f M × , (χM × + ϕ)(f) = χM × (f)+ ϕ(f) = ∈ = χM × (f) , ( + ϕ(f) = , otherwise. ∞ ∞ and (ϕ+χM × )(f)= χM × in the same manner. The supplement is clear by Lemma 1.4.13 and Remark 1.5.2.

Corollary 1.5.9. N- spec( ) is a contravariant functor from the category of binoids B into the − category of (boolean) semibinoids for every positive (boolean) binoid N.

Proof. This follows from Lemma 1.5.8. If ϕ : M M ′ is a morphism in the category of binoids, the → morphism hom(ϕ, N) : hom(M ′,N) hom(M,N) is given by ψ ψϕ. → 7→ Corollary 1.5.10. The canonical maps χ M ker χ and χ [ F are semibinoid isomorphisms 7→ \ F ← between 0, - spec M and ( (M) M , ,M ×) which are inverse to each other. { ∞} F \{ } ∩ Proof. The bijectivity is just a restatement of Proposition 1.4.41. The easy computations

ker(χ + χ′)= f M χ(f)+ χ′(f)= { ∈ | ∞} = f M χ(f)= f M χ′(f)= = ker χ ker χ′ { ∈ | ∞}∪{ ∈ | ∞} ∪ and 0 , if f F and f F ′ χF ∩F ′ (f)= ∈ ∈ = χF (f)+ χF ′ (f) = (χF + χF ′ )(f) ( , otherwise, ∞ prove the homomorphism property for both assignments.

Deﬁnition 1.5.11. The boolean binoid

v M := (hom(M, 0, ) χ , +,χ ,χ × ) { ∞} ∪{ M } M M is the d u a l and M vv is the bidual of M.

The bidual realizes the booleanization of ﬁnitely generated commutative binoids, cf. Corollary 4.2.10.

0 v Example 1.5.12. Let M be a binoid. If 0 is an adjoint identity element, then (M ) = ψ0 ψ v 0 0 v v { | ∈ M , where ψ0 is the extension of ψ to M . In particular, (M ) = M . } v ∼ 0 If M is a ﬁnitely generated free commutative binoid, then M = ( 0, n) for some n 1 by vv { ∞} ≥ Lemma 1.5.7, and hence M = 0, n. { ∞} vv Remark 1.5.13. There is a canonical binoid homomorphism δ : M M with f δf , where v →v 7→ δf : M 0, is the evaluation δf (χ) := χ(f) at f for every χ M . Note that ker δ = nil(M), →{ ∞} ∈ vv hence δf coincides with the absorbing element χ(M v )× : ϕ of M if and only if f nil(M). 7→ ∞ vv ×∈ Moreover, δ coincides with the identity element χ v : ϕ 0 of M if and only if f M . f M 7→ ∈ 28 1 Basic concepts of binoids

The Example 1.5.12 above shows that M vv = M happens. Later we will show that the homo- 6 vv morphism δ of the preceding remark is an isomorphism (i.e. M ∼= M) if and only if M is finitely generated and boolean, cf. Corollary 4.2.10. Hence, (M v )vv = M v for all finitely generated binoids M, cf. Corollary 4.2.11. In particular, P(V )vv = P(V ) if and only if V is infinite. So far, we have ∩ 6 ∩ the following examples.

Example 1.5.14. By Example 1.5.5(2), ( 0, n)v = 0, n, and hence ( 0, n)vv = 0, n. vv v vv { ∞}v { ∞} { ∞} { ∞} In particular, Pn,∩ =Pn,∩ =Pn,∩ and Pn,∪ =Pn,∪ =Pn,∪ by Example 1.2.4(1).

M v = M need not be true for every ﬁnitely generated boolean binoid as the following example shows.

Example 1.5.15. For the boolean binoid B = FC(b1,b2,b3)/(b1 + b2 = , 2bi = bi,i 1, 2, 3 ), v ∞ ∈ { } one has B = χ1,χ2,χ1,3,χ2,3 , where χI , I 1, 2, 3 , is the indicator homomorphism B 0, h i v ⊆{ } v →{ ∞} deﬁned by bi 0 for i I. Hence, B = B. If one denotes the generators of B by d1, d2, d3, and 7→ ∈ 6 vv d , then (with the same notation as above) B = χ ,χ ,χ = B. 4 ∼ h 1,2 1,3 2,4i ∼

1.6 Congruences

Homomorphisms on a monoid M and its homomorphic images are closely related to the notion of congruences on M, see [28, Chapter I.4], [29, Chapter I.3 and I.4], or [30, Chapter I.2]. This relation transfers to binoids since a congruence on a binoid is nothing else than a congruence on the underlying monoid, cf. Proposition 1.6.2. At the end of this section we prove Rédei’s Theorem for finitely generated commutative binoids. In the next section, we will give attention to specific congruences with which we will be concerned later.

Definition 1.6.1. A congruence on a binoid M is an equivalence relation on M that is compatible ∼ with the addition; that is, if a,b M with a b, then a + c b + c and c + a c + b for all c M. ∈ ∼ ∼ ∼ ∈ We denote the congruence class of a M by [a], sometimes also bya ¯, and the set of all congruence ∈ classes by M/ . Let and be two congruences on a binoid. The intersection is the ∼ ∼1 ∼2 ∼1 ∩∼2 congruence on M defined by a b for a,b M if a b and a b. We write if a b ∼ ∼ ∈ ∼1 ∼2 ∼1 ≤∼2 ∼1 implies a b for all a,b M . ∼2 ∈ On the set of congruences on a binoid, defines a partial order under which there always exist a ≤ largest and a smallest congruence, which means there are congruences and such that ∼u ∼id ∼id ≤ for all congruences on the binoid. These are given by the universal congruence , ∼≤∼u ∼ ∼u which relates any two elements with each other, and the identity congruence , under which two ∼id elements are related only when they coincide.

Before studying congruences on free commutative and ﬁnitely generated binoids, the equivalent notion of homomorphisms and congruences on a binoid is described in the following proposition.

Proposition 1.6.2. Let M be a binoid. Given a congruence on M, the quotient M/ is a binoid ∼ ∼ with respect to the operation [a] + [b] := [a + b] such that the canonical map M M/ , a [a], is → ∼ 7→ an epimorphism of binoids. Conversely, if ϕ : M N is a binoid epimorphism, then the relation → ∼ϕ on M deﬁned by a b if ϕ(a)= ϕ(b), is a congruence on M such that M/ N, [a] ϕ(a), is ∼ϕ ∼ϕ → 7→ a binoid isomorphism.

Proof. It is easily checked that M/ is a binoid with respect to the given operation and that ∼ ∼ϕ deﬁnes a congruence on M. The converse follows from the subsequent lemma. 1.6 Congruences 29

Lemma 1.6.3. If ϕ : M N is a binoid homomorphism and a congruence on M with , → ∼ ∼≤∼ϕ then there is a unique binoid homomorphism ϕ˜ such that the diagram

ϕ M / N ②< ②② π ②② ② ϕ˜ ②② M/ ∼ commutes. In particular, given two congruences and on M, the relation is equivalent ∼1 ∼2 ∼1 ≤∼2 to the fact that the map M/ M/ , [a] [a] , is a well-defined binoid epimomorphism. ∼1 → ∼2 1 7→ 2 Proof. Defineϕ ˜([a]) := ϕ(a). To show that this is well-defined assume that [a] = [b] for a,b M. ∈ By assumption, a b, and henceϕ ˜([a]) = ϕ(a) = ϕ(b)=ϕ ˜([b]). By definition,ϕ ˜ is a binoid ∼ϕ homomorphism withϕπ ˜ = ϕ, and hence unique. The situation of the supplement is expressed in the following diagram π2 M / M/ 2 t: ∼ tt π1 tt tt ϕ˜ tt M/ ∼1 with π2 =ϕπ ˜ 1 surjective. Hence,ϕ ˜ is surjective.

Remark 1.6.4. In standard literature, the congruence deﬁned by a monoid homomorphism ∼ϕ ϕ : M N as in Proposition 1.6.2 is usually denoted by ker ϕ, see for example [29], [30], or [49]. → Most probably this is due to the fact that the famous isomorphism theorem

M/ = im ϕ ∼ϕ ∼ can be stated with the common notation, cf. for instance [49, Proposition 8.2], and maybe because there is no absorbing element taken into account that justiﬁes our deﬁnition of ker. Note that (continuing with our notation) the binoid homomorphism induced by ker ϕ need not be an isomorphism, see

Remark 2.1.17 below. On the other hand, if ϕ is a binoid homomorphism such that ϕ|M\ker ϕ is injective, which means that ker ϕ is deﬁned by the congruence , then the induced homomorphism ∼ϕ is an isomorphism, cf. Remark 2.1.17. For instance, the congruences given in Lemma 1.7.6 below reﬂect this situation.

It is very common to identify a congruence with the set R( ) := (a,b) a b M M. For ∼ ∼ { | ∼ }⊆ × instance, R( )= M M and R( ) is the diagonal D := (a,a) a M M M. On the other ∼u × ∼id { | ∈ }⊆ × hand, every subset R M M generates a congruence on M. For this set ⊆ × R′ := (a + c,b + c) (a,b) R R−1 D,c M , { | ∈ ∪ ∪ ∈ } where R−1 = (b,a) (a,b) R . Now the relation on M given by { | ∈ } ∼R a b : a = a ,a ,...,a = b such that (a ,a ) R′ , i 1,...,n 1 , ∼R ⇔ ∃ 1 2 n i i+1 ∈ ∈{ − } is a congruence on M.

Definition 1.6.5. Let M be a binoid and R M M a subset. The congruence defined as ⊆ × ∼R above is the congruence generated by the relations R. A congruence on a binoid M is called ∼ finitely generated if there exists a finite subset R M M such that is the congruence ⊆ × ∼ generated by R. 30 1 Basic concepts of binoids

Remark 1.6.6. (Noetherian binoids) The set of all cogruences on a binoid M is a partially ordered set with respect to , where is equivalent to R( ) R( ). ≤ ∼1 ≤∼2 ∼1 ⊆ ∼2 In [28, Chapter 1.4], Gilmer defined a commutative monoid M to be noetherian if this order on the set of congruences on M is noetherian; that is to say, M satisfies the ascending chain condition (a.c.c.) on congruences, or in other words, for every chain of congruences ∼1 ≤ ∼2 ≤ ∼3 ≤ · · · there is a k N such that = for all n k. Equivalently, every congruence on M is finitely ∈ ∼n ∼k ≥ generated. Moreover, Gilmer showed that a commutative monoid is noetherian if and only if it is finitely generated, cf. [28, Theorem 5.10 and 7.8]. Since a binoid is finitely generated if and only if it is finitely generated as a monoid, and a congruence on a binoid is a congruence on the underlying monoid, we have the same statement for commutative binoids. Thus, considering noetherian or finitely generated commutative binoids amounts to the same thing. There is another (weaker) definition of noetherian in use by several authors in terms of ideals, cf. Remark 2.1.26.

Example 1.6.7. Let M be a commutative binoid generated by a i I . If ε : FC(I) M is the { i | ∈ } → canonical epimorphism i a , then M = FC(I)/ by Proposition 1.6.2. Thus, M is given by the 7→ i ∼ ∼ε generating set a i I and the family of relations ε(y )= ε(z ) (i.e. : y z , j J), which { i | ∈ } j j Rj j ∼ε j ∈ generate . This justifies the notation given in Definition 1.3.1, ∼ε M = FC(I)/( ) . Rj j∈J If #I = r, then M = (Nr)∞/ , where ε : e a , i 1,...,r . In Section 1.12, finitely generated ∼ ∼ε i 7→ i ∈{ } commutative binoids are studied in detail.

Similar to the situation for semigroups and monoids the index set J in the preceding example can be replaced by a finite subset if M is a finitely generated commutative binoid. This result is known as the Theorem of Rédei, and the proof for binoids is identical.

Lemma 1.6.8. (Preston) Let be a congruence on the free commutative binoid (Nn)∞ and I( ) the ∼ n ∞ ∼ ideal in the polynomial ring Z[X1,...,Xn], which is the binoid algebra of (N ) over Z, generated by the binomials Xa Xb, where a,b (Nn)∞ such that a b and Xc = Xc1 Xcn for c = − ∈ ∼ 1 ··· n (c ,...,c ) Nn and X∞ =0. Then a b if and only if Xa Xb I( ). 1 n ∈ ∼ − ∈ ∼ Proof. The proof of the statement for the monoid Nn can be found in [30, VI, Lemma 1.1] for instance. The argumentation for (Nn)∞ is exactly the same.

Theorem 1.6.9. (R´edei) Every congruence on the free commutative binoid (Nn)∞ is ﬁnitely generated.

Proof. Let be a congruence on (Nn)∞. If is not ﬁnitely generated, there exists an ascending chain ∼ ∼ R R R of ﬁnite subsets of R( ) = (a,b) a b , which induces an ascending 1 ⊂ 2 ⊂···⊂ k ⊂ · · · ∼ { | ∼ } chain of congruences

∼R1 ≤ ∼R2 ≤ ··· ≤∼Rk ≤ · · · on (Nn)∞. By assumption on , this chain cannot become stationary, and hence the same holds for ∼ the ascending chain of ideals

I( ) I( ) I( ) ∼R1 ⊆ ∼R2 ⊆ ··· ⊆ ∼Rk ⊆ · · · in Z[X1,...,Xn] by Lemma 1.6.8, contrary to Hilbert’s Basis Theorem, cf. [47, Theorem 1.C.4].

Similar to the theory of monoids, the following consequence of Theorem 1.6.9 may be called R´edei’s Theorem for ﬁnitely generated commutative binoids. 1.7 Important congruences 31

Corollary 1.6.10. Every finitely generated commutative binoid M with generators x1,...,xr admits finitely many relations ,..., such that M = FC(x ,...,x )/( , j 1,...,s ). R1 Rs ∼ 1 r Rj ∈{ } Proof. This follows immediately from Theorem 1.6.9 since M = (Nr)∞/ , where ε : (Nr)∞ M is ∼ ∼ε → the canonical binoid homomorphism e x , i 1,...,r . i 7→ i ∈{ } Example 1.6.11. Though finite, the number of relations on a finitely generated commutative binoid might be arbitrary large as the following example of binoids with only two generators shows:

M := FC(x, y)/( , k n,n +1,..., 2n 1 ) with : kx = ky , n Rk ∈{ − } Rk n 2. Every such binoid M is obviously positive, integral, and not torsion-free. Moreover, M is ≥ n n not cancellative since x is not cancellative as

x + ((n 1)x + y) = nx + y = ny + y = (n + 1)y = (n + 1)x = x + nx − but (n 1)x + y = nx. Similarly, y is not cancellative, which already implies that M contains no − 6 n non-trivial cancellative elements by Lemma 1.4.26.

Corollary 1.6.12. If M is a ﬁnitely generated or commutative binoid, then so is M/ . ∼ Proof. This is clear.

Of course, if is a congruence on a binoid M, not all properties of M transfer to M/ . In fact, the ∼ ∼ binoid M/ might be nicer than the initial binoid M if M = (Nr)∞, r 1. ∼ 6 ≥

1.7 Important congruences

In this section, we give a list of some congruences with which we will be concerned in this thesis. Most of them have a well-known counterpart in monoid theory, see for instance [28, Chapter I.4].

Definition 1.7.1. A congruence on M with the property that π : M M/ is injective on ∼ → ∼ M ker π is called an i d e a l c o n g r u e n c e . \ Example 1.7.2. Every binoid homomorphism ϕ : M N defines a (very important) ideal congruence → on M, namely a b if a = b or a,b ker ϕ. The elements of M/ are given by [a]= a if ∼ker ϕ ∈ ∼ker ϕ { } a ker ϕ and [a] = [ ] otherwise. In the following, we will make use of the notation 6∈ ∞ M/ ker ϕ for M/ . This notation will be justified in Section 2.1, where we will see that the congruence ∼ker ϕ defined by an ideal is an ideal congruence and every ideal congruence is up to isomorphism given by an ideal, cf. Remark 2.1.17.

Lemma 1.7.3. Every binoid homomorphism ϕ : M N factors uniquely through M/ ker ϕ such that → M/ ker ϕ is integral if N is so.

Proof. The factorizationϕ ˜ : M/ ker ϕ N follows from Lemma 1.6.3 and the supplement from the → fact thatϕ ˜(a)= is equivalent to a = . ∞ ∞ Lemma 1.7.4. If M is a positive binoid, then so is M/ for every ideal congruence . ∼ ∼ Proof. This is clear. 32 1 Basic concepts of binoids

Proposition 1.7.5. Let M be a binoid and an ideal congruence on M. If π : M M/ denotes ∼ → ∼ the canonical projection, then

N- spec(M/ ) = ϕ N- spec M ker π ker ϕ ∼ ∼ { ∈ | ⊆ } as semigroups. Proof. The canonical projection π : M M/ induces by Proposition 1.5.6 a semigroup homomor- → ∼ phism N-spec M/ N-spec M with ψ ψπ such that ψπ(a) = for all a π−1([ ]). On the ∼ → 7→ ∞ ∈ ∞ other hand, if ϕ N-spec M with ker π ker ϕ, then , and hence ϕ factors through a binoid ∈ ⊆ ∼≤∼ϕ homomorphismϕ ˜ : M/ N with ϕ =ϕπ ˜ by Lemma 1.6.3. This shows that N-spec(M/ ) ψ ∼ → ∼ →{ ∈ N-spec M ker π ker ϕ with ψ ψπ is surjective, and it is injective because φπ = ψπ implies | ⊆ } 7→ φ = ψ by the surjectivity of π.

Lemma 1.7.6. Let M be a commutative binoid. (1) The relation on M given by a b if a = b or a,b intc(M), is an ideal congruence such ∼int ∼int ∈ that M/ = M . ∼int ∼ int (2) The relation on M given by a b if a = b or a,b nil(M), is an ideal congruence such ∼red ∼red ∈ that M/ =: M . ∼red red Proof. Clearly, int and red are equivalence relations. To show that int is a congruence let a,b M ∼ ∼ ∼ c ∈ with a int b. Clearly, a = b implies a + c = b + c for all c M. So consider the case a,b int (M). ∼ • ∈ ∈ Then a + x = = b + y for some x, y M . By the commutativity of M, we obtain (a + c)+ x = ∞ ∈ = (b + c)+ y for all c M, which shows a + c b + c. The last assertion of (1) follows from ∞ ∈ ∼int the observation that [a] = [ ] if and only if a intc(M) and [a]= a otherwise. The poof of (2) is ∞ ∈ { } similar to that of (1).

Corollary 1.7.7. If M is a commutative and N a reduced binoid, then N- spec M ∼= N- spec Mred as semigroups. Proof. By Proposition 1.5.6, the canonical projection π : M M induces an injective semigroup → red homomorphism N-spec M N-spec M, ψ ψπ. For the surjectivity, we need to show that red → 7→ every binoid homomorphism ϕ : M N factors through π : M M ; that is, ϕ =ϕπ ˜ for a binoid → → red homomorphismϕ ˜ : M N. Since N is reduced, ϕ(a)= for every a nil(M) and ϕ N-spec M. → ∞N ∈ ∈ Hence . Now the surjectivity follows from Lemma 1.6.3. ∼red ≤∼ϕ Deﬁnition 1.7.8. Let M be a commutative binoid. Two elements a,b M are called a s s o c i a t e d ∈ if a = b + u for some unit u of M. Lemma 1.7.9. Let M be a commutative binoid. The relation on M given by a b if a,b are ∼pos ∼pos associated, is a congruence such that M/ =: M is a positive binoid. Moreover, M = M if ∼pos pos pos ∼ and only if M is positive. Proof. Clearly, is an equivalence relation. To show that it is a congruence let a b. Then ∼pos ∼pos a = b + u for some u M ×, which implies a + c = b + c + u for all c M by the commutativity of ∈ ∈ M. Hence, a + c b + c. The last assertion follows from the observation that [u] = [0] for every ∼pos unit u. The supplement is clear.

The composition of the canonical embedding ι and the canonical projection πpos

π (M ×)∞ ι M pos M −→ −→ pos yields π ι(u) = [0] for all u M ×. In general, M is not the product of (M ×)∞ and M , i.e. pos ∈ pos M = (M ×)∞ M . This already follows for M ﬁnite often from a counting argument. If #M × = k 6∼ × pos 1.7 Important congruences 33 and a = a+u for a = and u M × 0 (which holds for example if M is separated, cf. Section 5.1), 6 6 ∞ ∈ \{ } then the equivalence classes with respect to have all k elements with the exception of [ ]= . ∼pos ∞ {∞} If #Mpos = n+1, then M has kn+1 elements and not (k +1)(n+1) as the product. See also Remark 1.8.5 and Example 1.8.6. There are also congruences related to torsion-freeness, asymptotically equivalence, and cancellativity.

Lemma 1.7.10. Let M be a commutative binoid. (1) The relation on M given by a b if na = nb for some n 1, is a congruence such that ∼tf ∼tf ≥ M/ =: M is a torsion-free binoid. In particular, M = M if and only if M is torsion-free. ∼tf tf tf ∼ (2) The relation on M given by a b if a,b are asymptotically equivalent, is a congruence such ∼ae ∼ae that M/ is free of asymptotic torsion. ∼ae Proof. (1) Obviously, is reﬂexive and symmetric. If a b and b c, say na = nb and mb = mc ∼tf ∼tf ∼tf for n,m 1, then (nm)a = (nm)c, and hence is transitive. By the commutativity of M, the ≥ ∼tf relation is compatible with the operation on M since na = nb implies n(a + c) = na + nc = ∼tf nb + nc = n(b + c) for every c M. Finally, M/ is torsion-free because if n[a] = n[b], then ∈ ∼tf m(na)= m(nb) for some m 1, hence [a] = [b]. The poof of (2) is similar to that of (1). ≥ Lemma 1.7.11. Let M be a commutative binoid.

(1) If M is positive, then so is Mtf . (2) If N is a torsion-free binoid, then N- spec M ∼= N- spec Mtf as semigroups. Proof. The ﬁrst statement is obvious and the second follows similar to Corollary 1.7.7

Lemma 1.7.12. Let M be a commutative binoid and S a submonoid of M. The relation on ∼can,S M given by a b if a + s = b + s for some s S, is a congruence such that [s] is cancellative in ∼can,S ∈ M/ =: M for all s S. ∼can,S can,S ∈ Proof. Only the transitivity is not trivial. So assume that a b and b c for a,b,c M. ∼can,S ∼can,S ∈ Then a + s = b + s and b + t = c + t for some s,t S. Since M is commutative, this gives ∈ a + (s + t)= b + s + t = c + (s + t) with s + t S. Hence, a c. ∈ ∼can,S If S is a submonoid of M such that S, then all elements are congruent to each other, in which ∞ ∈ case Mcan,S = [ ] . Note that the binoid Mcan,S need not be positive if M is so. However, we will { ∞ } • prove later, cf. Lemma 5.1.14, that this is true if M contains no elements f M with f = f + g for ∈ some g = 0. 6

Deﬁnition 1.7.13. Let M be a commutative binoid. The binoid Mcan := Mcan,int M is called the cancellation (binoid) of M.

Proposition 1.7.14. Let M be a commutative and C a regular binoid. (1) Every binoid homomorphism ϕ : M C with ker ϕ int M = factors uniquely through M . → ∩ ∅ can (2) Every binoid homomorphism ϕ : M C has a unique factorization → ϕ M / C . ✆B ✆✆ π ✆✆ ✆✆ M/ ker ϕ ✆✆ ✆✆ ϕ˜ ✆✆ πcan ✆ ✆✆ ✆ (M/ ker ϕ)can 34 1 Basic concepts of binoids

Proof. The second statement follows from the first since the factorization through M/ ker ϕ satisfies the assumption of (1) by Lemma 1.7.3. By Lemma 1.6.3, it suffices to show that under the assumptions of (1) on ϕ : M C one has . So let ker ϕ int M = and assume that a b. → ∼can,int M ≤∼ϕ ∩ ∅ ∼can,int M This is equivalent to a + s = b + s for a,b M and s int M, which gives ∈ ∈ ϕ(a)+ ϕ(s)= ϕ(a + s)= ϕ(b + s)= ϕ(b)+ ϕ(s) .

If this is , then ϕ(a) = ϕ(b) = because of the integrality of C and ϕ(s) = by assumption. ∞ ∞ 6 ∞ In the other case, when ϕ(a)+ ϕ(s) = ϕ(b)+ ϕ(s) = , the cancellativity of C yields ϕ(a) = ϕ(b). 6 ∞ Hence, a b. ∼ϕ Remark 1.7.15. The preceding result may fail when the binoid C is cancellative but not integral. As an example consider M := FC(x, y)/(x + y =2x + y, 2y = ) and the binoid homomorphism ∞ ϕ : M N∞ (N∞/(2 = )) , −→ × ∞ deﬁned by x (1, 1) and y ( , 1). The two sets 7→ 7→ ∞ ker ϕ = nx + my n,m 1 and int M = nx n 1 { | ≥ } { | ≥ } are disjoint, but ϕ does not factor through M = M/(y = x + y)= FC(x, y)/(y = x + y, 2y = ). can ∞ Finally we apply these congruences to the product of a family of binoids.

Lemma 1.7.16. Let (M ) be a family of nonzero binoids and S M , i I, a family of sub- i i∈I i ⊆ i ∈ monoids. (1) M = int(M ) . i∈I i int i∈I i ∪ {∞Π} (2) Qi∈I Mipos = Qi∈I (Mi)pos.

(3) IfQI is ﬁnite, thenQ i∈I Mi tf = i∈I (Mi)tf .

(4) If S := i∈I Si, thenQ i∈I Mi canQ,S = i∈I (Mi)can,Si . Proof. The firstQ assertion followsQ from ExampleQ 1.4.4(5). (2) By definition, (a ) (b ) for i i∈I ∼pos i i∈I (a ) , (b ) M is equivalent to (a + u ) = (a ) + (u ) = (b ) + (u ) = i i∈I i i∈I ∈ i∈I i i i i∈I i i∈I i i∈I i i∈I i i∈I (b + u ) , where (u ) ( M )× = M ×, cf. Example 1.4.9, which is equivalent to i i i∈I Q i i∈I ∈ i∈I i i∈I i a b on M for all i I. Similarly, one proves (4). (3) It is clear that (a ) (b ) implies i ∼pos i i ∈ Q Q i i∈I ∼tf i i∈I a b on M for all i I. Conversely, if for all i I there is an n 1 such that n a = n b , then i ∼tf i i ∈ ∈ i ≥ i i i i m(a ) = m(b ) with m := n . Hence, (a ) (b ) in M . i i∈I i i∈I i∈I i i i∈I ∼tf i i∈I i∈I i Q Q Remark 1.7.17. (1) There is no similar result for . For instance, if N and M are binoids, a,a′ nil M, and ∼red ∈ b nil N, then (a,b) (a′,b) on M N, but a a′ on M and b b on N. 6∈ 6∼red × ∼red ∼red (2) Lemma 1.7.16(3) need not be true if I is infinite. As a counterexample consider a family (Mn)n∈N of binoids with a ,b M such that na = nb for all n N. Then a b on M but n n ∈ n n n ∈ n ∼tf n n (a ) N (b ) N on M . n n∈ 6∼tf n n∈ i∈I i In Section 2.1 and Section 4.2,Q we will encounter two more important congruences, namely the Rees congruence of an ideal, and the congruence that yields the booleanization of a binoid. The ∼I ∼bool next sections are devoted to important equivalences and congruences on the disjoint union and the product of binoids. 1.8 Smash product 35

1.8 Smash product

Similar to monoid theory, we have shown in Section 1.1 that the product of a family of binoids is again a binoid and that the direct sum coincides with the product if and only if the family is finite. This and the next section is devoted to other constructions for binoids which have no counterpart in the theory of monoids, but which yield particularly interesting binoid algebras. As is well-known, the coproduct in the category of commutative monoids is the direct sum, whereas the (finite) coproduct in the category of arbitrary monoids is given by the free monoid on the disjoint union of the monoids modulo a particular equivalence relation that ensures the universal property, cf. [5, Example 3.9]. The same phenomenon occurs when dealing with binoids. For this reason, we only focus on the category of commutative binoids when determining the coproduct. Nonetheless, the construction of the coproduct makes sense for arbitrary binoids (as the direct sum for monoids) and for pointed sets in general, which we will call the smash product.2 Moreover, several examples show that the binoid algebra (over a ring R) of the smash product of binoids yields the tensor product (over R) of the binoid algebras, which will be proved later in Corollary 3.5.2. In the next section, when we introduce the analogue of modules and algebras for binoids, namely N - sets and N - binoids, we will extend the definition of the smash product. For this, we need to start here with pointed sets in general. Definition 1.8.1. Let (S ,p ) be a family of pointed sets and denote by the relation on S i i i∈I ∼∧ i∈I i given by Q (s ) (t ) : s = t , i I, or s = p ,t = p for some k,ℓ I. i i∈I ∼∧ i i∈I ⇔ i i ∀ ∈ k k ℓ ℓ ∈ Then the pointed set S := S i i ∼∧ i^∈I Yi∈I . with distinguished point [(pi)i∈I ] =: p∧ is called the s m a s h p r o d u c t of the family (Si,pi)i∈I . The class [(s ) ] S for some (s ) S will be denoted by s . i i∈I ∈ i∈I i i i∈I ∈ i∈I i ∧i∈I i Let (Mi)i∈I beV a family of binoids. ConsideringQ every binoid as a pointed set with distinguished element , the relation on M identifies all tuples with at least one entry equal to with ∞i ∼∧ i∈I i ∞i the absorbing element ( ) = and leaves the rest untouched. ∞i i∈I Q∞Π Example 1.8.2. The binoid M = FC(x, y)/(3x = x, 2y = 0) is the smash product of (M ×)∞ and M , i.e. M = (M ×)∞ M . Moreover, for a ring K we get pos ∼ ∧ pos K[M]= K[X, Y ]/(X3 X, Y 2 1) − − = K[X]/(X3 X) K[Y ]/(Y 2 1) − ⊗K − = K[M ] K[(M ×)∞] . pos ⊗K Lemma 1.8.3. Given a family (M ) of binoids, the relation on M is an ideal congruence i i∈I ∼∧ i∈I i such that M is a binoid with identity element 0 =: 0 and absorbing element =: i∈I i ∧i∈I i ∧ Q ∧i∈I ∞i . Moreover, the canonical inclusions ∞∧ V ι : M M , a a , k k −→ i 7−→ ∧i∈I i i^∈I where a = a and a =0 for i = k, are binoid homomorphisms. k i 6 Proof. All assertions are easily verified.

2 The terminology stems from algebraic topology. 36 1 Basic concepts of binoids

Note that the smash product is the zero binoid if one binoid of the family is so, and 0, = i∈I { ∞} ∼ 0, . In general, one has M 0, = M and M = for every binoid M. Thus, { ∞} ∧{ ∞} ∧ {∞} {∞∧} V 0, is an ‘identity object’ and an ‘absorbing object’ in the category of binoids B with respect { ∞} {∞} to the smash product.

Example 1.8.4. If (M ) is a ﬁnite family of nonzero binoids and A M a generating set of M , i i∈I i ⊆ i i i I, then M is generated by ∈ i∈I i V ae := 0 0 a 0 0 , i ∧···∧ ∧ ∧ ∧···∧ a A , i I, where a is the ith entry of ae . In particular, the generators of r N∞ are ∈ i ∈ b i i=1 e := 0 0 1 0 0 , V i ∧···∧b ∧ ∧ ∧···∧ i 1,...,n , where 1 is the ith entry of e . Note that r N∞ = (Nr)∞. ∈{ } b i i=1 ∼ Remark 1.8.5. Let G be a binoid group and N a positiveV binoid. Then G N is a binoid whose unit b ∧ group is G× and such that (G N) is isomorphic to N. In particular, there is an injective and a ∧ pos surjective binoid homomorphism (G×)∞ G N N −→ ∧ −→ such that their composition is just the characterisic point χG× .

Example 1.8.6. Consider the binoid group G = (Z/2Z)∞ = FC(y)/(2y = 0) and the binoid N = FC(x)/(3x = x). Their smash product is

G N = FC(x, y)/(3x = x, 2y = 0) . ∧ Thus, G N is the binoid from Example 1.8.2. Now consider the binoid ∧ M˜ = FC(x, y)/(3x = x + y, 2y = 0) .

Both binoids, G N and M˜ , have the same unit group 0,y = Z/2Z and coincide as sets, more ∧ { } ∼ precisely, one has G N = , 0, x, y, x+y, 2x, 2x+y = M˜ . Hence, (G N) = [ ], [0], [x], 2[x] = ∧ {∞ } ∧ pos { ∞ } N = M˜ pos as binoids. Thus,

((G N)×)∞ (G N) = (M˜ ×)∞ M˜ . ∧ × ∧ pos × pos However, G N = M˜ since 3a = a for all a G N, but 3(x + y) = x in M˜ . The corresponding ∧ 6∼ ∈ ∧ binoid algebras over a ring K

K[G N]= K[X, Y ]/(X3 X, Y 2 1) ∧ − − and K[M˜ ]= K[X, Y ]/(X3 YX,Y 2 1) − − are not isomorphic either.

Lemma 1.8.7. Let (Mi)i∈I be a family of nonzero binoids and N a binoid. Every binoid homomorphism ϕ : M N with ϕ((a ) ) = if a = for at least one i I, factors through i∈I i → i i∈I ∞ i ∞ ∈ M . i∈I i Q VProof. This is an immediate consequence of Lemma 1.6.3 since , where π denotes the canonical ∼ϕ≤∼π epimorphism M M . i∈I i → i∈I i Q V 1.8 Smash product 37

Lemma 1.8.8. Let (Mi)i∈I be a family of nonzero binoids.

(1) i∈I Mi is commutative if and only if all Mi are commutative. × × (2) V i∈I Mi ∼= i∈I Mi . (3) int M = int(M ) and intc M = a a intc(M ) for at least one V i∈I i ∼Q i∈I i i∈I i {∧i∈I i | k ∈ k k I . In particular, i∈I Mi int = i∈I (Mi)int; that is, i∈I Mi is integral if and only if all ∈V} Q V • M are integral. In this case, M = ( M )∞. i V i∈I i V i∈I i V (4) nil M = a a nil(M ) for at least one k I . i∈I i {∧i∈I i | k ∈V k Q ∈ } (5) i∈VI Mi pos = i∈I (Mi)pos.

(6) IfVI is finite, thenV i∈I Mi tf = i∈I (Mi)tf . (7) Let S M , i I, be a family of submonoids. If S := s s S M , then i ⊆ i ∈ V V {∧i∈I i | i ∈ i} ⊆ i∈I i M = (M ) . i∈I i can,S i∈I i can,Si V Proof.VBy definition, theV class of an element in M under is a singleton if (and only if) there i∈I i ∼∧ is no entry equal to and otherwise. In particular, is an ideal congruence. Having observed ∞i ∞∧ Q ∼∧ this, most statements follow by comparsion with similar statements on the product i∈I Mi. See, for instance, Lemma 1.7.16 for (5)-(7). The first statement is obvious by Lemma 1.1.10. The statements Q (2)-(4) are easily verified. See also Example 1.4.4(5).

Remark 1.8.9. As for the product, cf. Remark 1.7.17(1), there are no such results like (5)-(7) for the congruence on the smash product. For instance, since [e ] = [e ]= in M , one ∼red i,∞ j,∞ ∞∧ ∧i∈I i has [e ] [e ] but 0 on every nonzero binoid. i,∞ ∼red j,∞ 6∼red ∞ The smash product is a universal object, namely, it is the (ﬁnite) coproduct in the category of commutative binoids comB. This is contained in the following result.

Proposition 1.8.10. Let (Mi)i∈I be a finite family of binoids and N a commutative binoid. Every family ϕ : M N, i I, of binoid homomorphisms gives rise to a unique binoid homomorphism i i → ∈ , ϕ : M N such that ϕι = ϕ , where ι denotes the canonical embedding M ֒ M i∈I i → k k k k → i∈I i a ae , k I. 7→V k ∈ V Proof. Define ϕ( a ) := ϕ (a ) for a M . It is easily checked that this is a b ∧i∈I i i∈I i i ∧i∈I i ∈ i∈I i well-defined binoid homomorphism with ϕ = ϕι for all k I. P k k V ∈ Similar to the situation in the category of monoids, the induced map ϕ in the proof of Corollary 1.8.10 is a homomorphism because N is commutative.

n ∞ n ∞ n ∞ n Example 1.8.11. Since i=1 N = (N ) , one has N-spec i=1 N = N by Lemma 1.5.7, which yields by Example 1.5.12 V V

n v n vv 0 N∞ = ( 0, n) and N∞ = 0, n . { ∞} { ∞} i=1 i=1 ^ ^

Proposition 1.8.12. Let (Mi)i∈I be a ﬁnite family of nonzero binoids. If N is a commutative binoid, then N- spec Mi ∼= N- spec Mi i^∈I Yi∈I as semigroups. The isomorphism is given by ϕ (ϕι ) , where ι denotes the canonical embedding 7→ i i∈I k .( M ֒ M , a ae , with inverse ϕ [ (ϕ ) , where ϕ( a ) := ϕ (a k → i∈I i 7→ i ← i i∈I ∧i∈I i ∈I i i Proof. VThe canonical embeddings ι : M M , k I, with a Pae induce by Proposition b k k → i∈I i ∈ 7→ i 1.5.6 a semigroup homomorphism N-spec M N-spec M with ϕ ϕι . By the universal i∈VI i → k 7→ k b V 38 1 Basic concepts of binoids property of the product in the category of semigroups, this gives rise to a semigroup homomorphism

ψ : N-spec M N-spec M i −→ i i^∈I Yi∈I with ϕ (ϕι ) , which is surjective by Proposition 1.8.10. For the injectivity suppose that ϕ, 7→ i i∈I ϕ′ N-spec M with ϕι = ϕ′ι for all k I. Since every element a M can be ∈ i∈I i k k ∈ ∧i∈I i ∈ i∈I i written as a e = ι (a ), we get iV∈I i i i∈I i i V

P P ′ ′ b ϕ( i∈I ai)= ϕιi(ai)= ϕ ιi(ai)= ϕ ( i∈I ai) . ∧ ∧ Xi∈I Xi∈I Hence, ϕ = ϕ′. For the supplement, one easily checks that ϕ as given in the statement is the preimage of (ϕi)i∈I under ψ.

To determine the N - spectrum of the product of a family of nonzero binoids, we need the following deﬁnition.

Deﬁnition 1.8.13. The disjoint union ofafamily (Si)i∈I of arbitrary sets is given by the set

S := (s; i) s S ,i I . i { | ∈ i ∈ } i]∈I

Proposition 1.8.14. Let (Mi)i∈I be a ﬁnite family of nonzero binoids. If N is a commutative binoid with no non-trivial idempotents, then

N- spec Mi ∼= N- spec Mi ∼= N- spec Mi Yi∈I ∅6=]J⊆I i^∈J ∅6=]J⊆I iY∈J as semigroups, where the semigroup structure in the middle and on the right-hand side are given by

(ψ; J) + (ψ′; J ′) := (ψ + ψ′; J J ′) , ∪ ′ ′ where (ψ + ψ )((ai)i∈J∪J′ ) := ψ((ai)i∈J )+ ψ ((ai)i∈J′ ), and

′ ′ ′ ′ ((ψ ) ; J) + ((ψ ) ; J ) := ((ψ ) + (ψ ) ′ ); J J ) , i i∈J i i∈J i i∈J i i∈J ∪ ′ ′ where ((ψi)i∈J + (ψi)i∈J′ )((ai)i∈J∪J′ ) := (ψi(ai))i∈J + (ψi(ai))i∈J′ , respectively.

Proof. The latter semigroup isomorphism is an immediate consequence of Proposition 1.8.12 and given by (ψ; J) ((ψι ) ; J), where ι denotes the binoid embedding M ֒ M . To obtain the ﬁrst 7→ i i∈J i i → j∈J j isomorphism observe that for every J I, J = , the natural binoid epimorphism ππ , where ⊆ 6 ∅ V J π π M J M M , i −→ i −→ i Yi∈I iY∈J i^∈J induces a semigroup embedding

ϕ : N-spec M N-spec M J i −→ i i^∈J Yi∈I with ψ ψππ . By the universal property of the disjoined union, these embeddings give rise to a 7→ J map ϕ : N-spec M N-spec M i −→ i ∅6=]J⊆I i^∈J Yi∈I 1.8 Smash product 39 with (ψ; J) ψππ . It is easily checked that this is a semigroup homomorphism, where the disjoint 7→ J union is a semigroup as described in the proposition. We show that ϕ is a bijection. For this take an arbitrary φ N-spec M and set ∈ i∈I i Q J := i I φ(e )= . { ∈ | i,∞ ∞} By Lemma 1.4.39(2) and the assumption on N, we obtain φ(e ) bool(N) = 0, . Since i,∞ ∈ { ∞} e = , we have φ(e )= in N, which therefore implies that φ(e )= for at least i∈I i,∞ ∞ i∈I i,∞ ∞ i,∞ ∞ one i I. In particular, J = . We claim that φ factors through M . For this, we may assume P ∈ P6 ∅ i∈J i that I = 1,...,n and that J = 1,...,k . Since 0 = φ( e ) (= φ(0,..., 0, ,..., )), we { } { } i6∈J Qi,∞ ∞ ∞ obtain P φ(a ,...,a ) = φ (a ,...,a )+ e = φ(a ,...,a , ,..., ) , 1 n 1 n i,∞ 1 k ∞ ∞ Xi6∈J which shows that φ depends only on the J -components. Therefore, φ factors through the binoid homomorphism

φ˜ : M N, (a ,...,a ) φ(a ,...,a , ,..., ) . i −→ 1 k 7−→ 1 k ∞ ∞ iY∈J Hence, φ˜ factors through M by Lemma 1.8.7. This shows the surjectivity of ϕ. The injectivity ∧i∈J i is clear since J is uniquely determined by φ.

The condition on N containing only trivial idempotent elements is necessary in the preceding proposition. Consider, for instance, the binoid M = FC(x , x )/(x + x = , 2x = x , 2x = x ). The 1 2 1 2 ∞ 1 1 2 2 binoid homomorphism N∞ N∞ M × −→ with e 0 and e x , i 1, 2 , does not factor through one of the factors. i 7→ i,∞ 7→ i ∈{ }

Example 1.8.15. We apply Proposition 1.8.14 to determine the N - spectra of the products 0, n { ∞} and (N∞)n, where N denotes a binoid with bool(N) = 0, . For this, recall that M = { ∞} i∈I i ( M )∞ if all M are integral, cf. Lemma 1.8.8(3), and let I = 1,...,n . Since 0, = i∈I i i { } i∈VI { ∞} ∼ 0, and N-spec 0, = 0, N = , we get {Q ∞} { ∞} {{ ∞} → } ∼ {∞} V N-spec 0, n = , { ∞} ∼ {∞} ∅6=]J⊆I as semigroups. Similarly, since r N∞ = (Nr)∞ for every r 1, we obtain i=1 ≥ V∞ n #J ∞ #J N- spec(N ) ∼= N-spec(N ) ∼= N ∅6=]J⊆I ∅6=]J⊆I as semigroups, where the latter isomorphism is due to Lemma 1.5.7. In particular, for the dual of 0, n and (N∞)n, we get { ∞} v v ( 0, n) = and ((N∞)n) = 0, #J { ∞} ∼ {∞} ∼ { ∞} J]⊆I J]⊆I See also Example 1.5.5(2). 40 1 Basic concepts of binoids

1.9 Pointed unions

In this section, we construct a binoid, called the bipointed union, from the disjoint union of a family of (positive) binoids, which was introduced at the end of the last section. As a first step, we define the pointed union of a family of pointed sets. Lemma 1.9.1. Given a family (S ,p ) of pointed sets, the relation on the disjoint union i i i∈I ∼∞ S that glues all points (p ; i), i I, together and leaves the rest untouched, i.e. i∈I i i ∈ U (s; i) (t; j): i = j and s = t or a = p and b = p , ∼∞ ⇔ i j defines an equivalence relation such that

S =: S i ∼∞ · i i]∈I . i[∈I is again a pointed set ( S ,p), where p = [(p ; i)], i I.3 In particular, if (M ) is a family of · i∈I i i ∈ i i∈I binoids (or semibinoids), then the addition S (a + b; i) , if i = j, (a; i) + (b; j) := ( , otherwise, ∞ deﬁnes a semibinoid structure on M . · i∈I i Proof. This is easily veriﬁed. S

Definition 1.9.2. Let (S ) be a family of pointed sets. The pointed set S is called the i i∈I · i∈I i p o i n t e d u n i o n . By the pointed union of a family (M ) of binoids, we always mean the i i∈I S semibinoid M with addition given as in Lemma 1.9.1. · i∈I i Note that S M contains no integral elements. There are examples where the pointed union of · i∈I i pointed sets turns into a binoid with respect to a certain addition different from the one of the pointed S union of binoids, cf. Remark 3.4.8. Remark 1.9.3. The disjoint union of a family (M ) , #I 2, of binoids turns not into a monoid i i∈I ≥ by glueing together the identity elements using the same construction as in the lemma above with 0 instead of everywhere. In fact, the operation (a; i) + (b; j) := (a + b; i) if i = j and 0 := [(0; i)] ∞ otherwise, is not associative since for i = j and a = 0 , one has ((a; i) + (b; i)) + (c; j) = 0 and 6 6 i (a; i) + ((b; i) + (c; j)) = (a; i) = 0. 6 For the same reason, the semibinoid M turns not into a binoid by glueing the identity elements · i∈I i (0; i), i I, together (so that 0 := [(0; i)] becomes the desired identity element) if at least one M is ∈ S i not positive. To see this let 0 = a M × and = b M , j = i. Then ((a; i)+(-a; i))+(b; j) = (b; j) i 6 ∈ i ∞j 6 ∈ j 6 and (a; i) + ((-a; i) + (b; j)) = ; that is, the operation on M is not associative anymore. ∞ · i∈I i There are other relations on the disjoint union of not necessarily positive binoids, where among other S identifications all identity elements are glued together as well as all absorbing elements, such that the quotient is a binoid with respect to a certain addition different from those considered in this section, cf. Lemma 1.11.2. Lemma 1.9.4. Let (M ) be a family of positive binoids. The relation on the disjoint union i i∈I ∼• i∈I Mi of the underlying sets of the binoids given by

U (a; i) (b; j): i = j and a = b or a = and b = or a =0 and b =0 , ∼• ⇔ ∞i ∞j i j 3 Apart from the class of p, all classes are singletons and will therefore be written as (a; i) for a ∈ Si, i ∈ I, by abuse of notation. 1.9 Pointed unions 41 deﬁnes an equivalence relation such that

M =: : M i ∼• i i]∈I . i[∈I equipped with the addition

(a + b; i) , if i = j, (b; j) , if a =0, (a; i) + (b; j) :=  (a; i) , if b =0,  , otherwise, ∞   is a binoid with absorbing element := [( ; i)] and identity element 0 := [(0 ; i)]. Moreover, the ∞ ∞i i canonical inclusions ι : M : M with a (a; k) k k −→ i 7−→ i[∈I and projections a , if i = k , πk : : Mi Mk with (a; i) −→ 7−→ ( k , otherwise, i[∈I ∞ k I, are binoid homomorphisms. ∈ Proof. This is easy to check.

By deﬁnition, we have : M = M , where is the equivalence relation on M i∈I i · i∈I i ∼ ∼ · i∈I i that glues the identity elements (0; i), i I, together. S S∈ S

Deﬁnition 1.9.5. Let (Mi)i∈I be a family of positive binoids. With the notation of Lemma 1.9.4, : the binoid i∈I Mi is called the b i p o i n t e d u n i o n of the family (Mi)i∈I . The bipointedS union of a family of positive binoids can be visualized as a family of lines, where each line represents a binoid M , i I, which are glued together at the ends. i ∈

∞ Mi 0

Note that a trivial binoid does not contribute to the bipointed union since M : 0, = M. ∪{ ∞} ∼

Remark 1.9.6. If (Mi)i∈I is a family of positive binoids, then

: M = M , i ∼ i ∼ i[∈I i^∈I . where denotes the ideal congruence on M given by a if a = 0 for at least two ∼ i∈I i ∧i∈I i ∼ ∞∧ i 6 i I. Thus, there are canonical binoid epimorphisms ∈ V π π∧ : M M : M . i −→ i −→∪ i Yi∈I i^∈I i[∈I 42 1 Basic concepts of binoids

Lemma 1.9.7. Let (Mi)i∈I be a family of positive binoids. : (1) The binoid i∈I Mi is commutative if and only if all Mi are commutative. (2) If at least two of the binoids are non-trivial, then int : M = 0 . In particular, : M S i∈I i { } i∈I i is a positive binoid in which all elements =0 are non-integral. 6 S S : : : : (3) nil i∈I Mi = i∈I nil(Mi) and i∈I Mi red = i∈I (Mi)red . : : : : (4) i∈SI Mi pos = Si∈I Mi and i∈ISMi tf = i∈I (SMi)tf . Proof. S(1)-(3) are immediate.S The ﬁrst S assertion of (4)S follows from (2) and Lemma 1.7.9. The latter is due to the observation that for a M and b M the equality n(a; i)= n(b; j) for some n 1 is ∈ i ∈ j ≥ equivalent to i = j and na = nb for some n 1 or a = b 0, . ≥ ∈{ ∞} Lemma 1.9.8. Let (M ) be a ﬁnite family of positive binoids. If A M is a generating set of i i∈I i ⊆ i M , i I, then : M is generated by (a; i), a A , i I. i ∈ i∈I i ∈ i ∈ Proof. This is obvious.S

The bipointed union admits a universal property.

Proposition 1.9.9. Let (Mi)i∈I be a ﬁnite family of positive binoids and N another binoid. Given a family of binoid homomorphisms ϕ : M N, i I, such that ϕ (a)+ ϕ (b)= for all a =0 and i i → ∈ i j ∞ 6 i b =0 , i = j, there is a unique binoid homomorphism 6 j 6 ϕ : : M N i −→ i[∈I with ϕι = ϕ , where ι : M : M denotes the canonical embedding a (a; i), i I. i i i i → i∈I i 7→ ∈ Proof. The unique binoid homomorphismS is obviously given by ϕ(a; i) := ϕ (a) for a M , i I. i ∈ i ∈

Corollary 1.9.10. Let (Mi)i∈I be a ﬁnite family of positive binoids and N an integral binoid. Then we have a semibinoid isomorphism:

N- spec : M = N- spec M . i ∼ · i i[∈I i[∈I : Proof. Set M := i∈I Mi. Note that the characteristic functions χM × : Mi N, i I, and i → ∈ χ × : M N are binoid homomorphisms by Proposition 1.4.41. In particular, N-spec M , i I, M → S i ∈ and N-spec M are semibinoids such that the pointed union is well-deﬁned, cf. Lemma 1.9.1. The canonical projections π : M M , k I, induce semigroup homomorphisms k → k ∈ N-spec M N-spec M, k −→ k I, which by the universal property of the disjoint union give rise to a map ∈ ϕ : N-spec M N-spec M. i −→ i]∈I This map factors through φ : N-spec M N-spec M, · i −→ i[∈I (ψ; i) ψπi, because ϕ((χM × ; i)) = χM × πi = χM × for all i I. It is easily checked that φ is 7→ i i ∈ a semibinoid homomorphism, which is obviously injective. For the surjectivity let ψ N-spec M ∈ (i.e. ψ : : M N). If ψ = χ × , then ψ is the image of the absorbing element under φ. So i∈I i → M S 1.9 Pointed unions 43

let ψ = χ × . Then there is a (a; k) M 0 with ψ((a; k)) = . In this case, we have for all 6 M ∈ \{ } 6 ∞ b M 0 , where i = k, ∈ i \{ } 6 = ψ( )= ψ((a; k) + (b; i)) = ψ(a; k)+ ψ(b; i) , ∞ ∞ which implies that ψ(b; i)= by the integrity of N. Hence, φ : ψι ψ. ∞ k 7→ : ∞ Example 1.9.11. We will double-check Corollary 1.9.10 by determining N-spec i∈I N step by step for an integral binoid N and I = 1,...,n . A generating set of the bipointed union : N∞ { } S i∈I is given by (1; i) i I , where the generators satisfy (1; i) + (1; j) = for all i, j I when { | ∈ } ∞ ∈S i = j. Since N is integral, at most one generator lies not in the kernel of any binoid homomorphism 6 : N∞ N. If for a N, i∈I → ∈ (ϕ ; k): : N∞ N S a −→ i[∈I denotes the binoid homomorphism (1; i) a if i = k and otherwise, then (ϕ ; i) = (ϕ ; j) =: α 7→ ∞ ∞ ∞ for all i, j I and ∈ (ϕa+b; i) , if i = j , (ϕa; i) + (ϕb; j) = (α , otherwise, From this we obtain

• N-spec : N∞ = (ϕ ; k) a N , k I α , +, α = N. { a | ∈ ∈ }∪{ } ∼ · i∈I i∈I [ [ : ∞ In particular, the dual of i∈I N is given by S 0 0, = : FC(x)/(2x = x) =: B. · { ∞} ∼ i[∈I i[∈I vv : ∞ Now the same argumentation yields i∈I N ∼= B. Example 1.9.12. To illustrate the diﬀerences S of the constructions we have encountered so far consider the following binoids: N∞ N∞ = N∞ N∞: N∞ N∞ = (N N)∞ : N∞ : N∞ : × ⊕ ∧ × ∪

(0, ∞) (∞, ∞) (∞, ∞) ∞

(0, 0) (∞, 0) (0, 0) 0

In terms of generators, these binoids are given by

FC(x,y,u,v)/(u + v = , x + u = u,y + v = v) , FC(x, y) , FC(x, y)/(x + y = ) , ∞ ∞ with binoid algebras4

K[N∞ N∞] = K[X] K[Y ] K[X, Y ] , × ∼ × × K[N∞ N∞] = K[X, Y ] , ∧ ∼ K[N∞ : N∞] = K[X, Y ]/(XY ) , ∪ ∼ 4 Here K[X] × K[Y ] × K[X,Y ] is the product in the category of rings. 44 1 Basic concepts of binoids over the ring K. The ﬁrst isomorphism is given by T (1,0) X and T (0,1) Y , and the last by 7→ 7→ T (1;1) X and T (1;2) Y . The result for the smash product follows from Lemma 1.8.8 and the 7→ 7→ theory of monoid rings since K[N∞ N∞]= K[(N N)∞]= K(N N)= KN KN = K[X, Y ]. ∧ × × ⊗K ∼ Example 1.9.13. We have the following R - spectra:

∞ ∞ ∞ ∞ ∞ R- spec(N ∪: N ) R- spec(N ∪: N ∪: N )

1.10 Operations

This section deals with pointed sets and binoids on which a binoid N operates. These so-called N - sets and N - binoids represent the binoid theoretic counterpart of modules and algebras over a ring. In this spirit, we extend the deﬁnition of the smash product to N - sets and N - binoids. In Section 3.4 and Section 3.5, we will discuss the associated modules and algebras of N - sets and N - binoids, respectively. A thorough investigation of N - sets and their modules can be found in [14, Section 2.2], where the notion of ﬁnitely generated, noetherian, and projective N - sets has also been introduced and studied in detail, while here we omit a treatment of the latter two.

Convention. In this section, N always denotes an arbitrary binoid.

Deﬁnition 1.10.1. A (left) operation of N on a pointed set (S,p) is a map

+ : N S S, (a,s) a + s , × −→ 7−→ such that the following conditions are fulﬁlled. (1) 0+ s = s for all s S. ∈ (2) + s = p for all s S. ∞ ∈ (3) a + p = p for all a N. ∈ (4) (a + b)+ s = a + (b + s) for all a,b N and s S. ∈ ∈ Then S is called an N - set. An N - m a p is a pointed map ϕ : S T of N - sets such that the → diagram N S / S × id ×ϕ ϕ N T / T × commutes; that is, ϕ(a + s)= a + ϕ(s) for all a N and s S. We say S isa finitely generated ∈ ∈ N - set if there exists a ﬁnite subset T S such that every s S can be written as s = a + t for some ⊆ ∈ a N and t T , i.e. S = (N + t). Then (S,p)is generated asan N -set by T . ∈ ∈ t∈T S 1.10 Operations 45

The addition M M M, (x, y) x + y, on a binoid M defines for every subbinoid N M an × → 7→ ⊆ operation on M by restricting the first component to N. In particular, M is an M - binoid. In general, if S is an N -set and N ′ N a subbinoid, then S is also an N ′ - set. The zero binoid operates only ⊆ on S = p , whereas the trivial binoid operates on every pointed set in the trivial way. { } Remark 1.10.2. Property (3) of the definition of an N - set says that p is an invariant element under the operation of N. Of course, p need not be unique with this property. Arbitrary sets with an operation (also called action) of a semigroup or monoid S are usually called S - acts. A thorough investigation of S -acts can be found in [36]. See also [45] for a treatment of the associated modules. However, when it comes to study properties and results of ring and module theory for S - acts, it is not uncommon to focus on S - acts that admit a unique invariant element and where S contains an absorbing element, which resembles our setting of N - sets. In [2], for instance, S - acts that satisfy some versions of Nakayama’s lemma and Krull’s (intersection) theorem are studied, which do not translate directly from rings to binoids either, cf. Remark 2.1.26. In general, there is a close connection between operations of an object as defined above and its representations, see the subsequent lemma. Taking up this point of view, S -acts are studied by Clifford and Preston in [15, Chapter 11] and called operands over S. Right from the beginning they frequently point out that there is no loss of generality in dealing with acts that admit a unique invariant element (centered operands) over a semigroup S with an absorbing element, which they do “almost exclusively”. Lemma 1.10.3. A pointed set (S,p) is an N - set if and only if there is a binoid homomorphism

N (map S, , id, ϕ ) , a ϕ , −→ p ◦ ∞ 7−→ a where ϕ : S S denotes the translation s a + s by a N. a → 7→ ∈ Proof. This is easy to check.

An N∞ - set is by Lemma 1.10.3 the same as S together with a ﬁxed pointed map ϕ : S S, the → operation being given by n + s = ϕn(s). Lemma 1.10.4. The product, the direct sum, the smash product, and the pointed union of a family of N - sets are again N - sets.

Proof. Let (Si,pi)i∈I be a family of N -sets. The operation of N on i∈I Si is given by a + (si)i∈I = (a + s ) , which immediately yields the operation of N on S and S . The operation of i i∈I i∈I Qi i∈I i N on S is given by a + (s; i) = (a + s; i). · i∈I i L V

DeﬁnitionS 1.10.5. Let (Si,pi)i∈I be a ﬁnite family of N -sets. If ∧ denotes the equivalence ∼ N relation on i∈I Si generated by V (a + si) sj ∧ si (a + sj ) , ···∧ ∧···∧ ∧··· ∼ N ···∧ ∧···∧ ∧ · · · where a N and s S , k I, then ∈ k ∈ k ∈

Si := Si ∧ N ∼ N i^∈I i^∈I . is called the s m a s h p r o d u c t of the family (S ) over N. A class [ s ] S will be i i∈I ∧i∈I i ∈ N,i∈I i denoted by s . Unless there is confusion, we will sometimes omit the index set and simply write ∧N,i∈I i V S and s . N i ∧N i

LemmaV 1.10.6. The smash product N Si of a ﬁnite family (Si,pi)i∈I of N - sets is again an N -set with distinguished point p =: p . If one component of s equals p , then s = p . ∧N i ∧N V ∧N i i ∧N i ∧N 46 1 Basic concepts of binoids

Proof. Let I = 1,...,n . It is clear that ( S ,p ) is a pointed set and that the map N S { } N i ∧N × N i → S with (a, s ) (a + s ) s s is a well-deﬁned operation of N on S . The N i ∧N i 7→ 1 ∧N 2 ∧NV···∧N n NV i supplement follows from V V p s s = (p + ) s s 1 ∧N 2 ∧N ···∧N n 1 ∞ ∧N 2 ∧N ···∧N n = p (s + ) s s 1 ∧N 2 ∞ ∧N 3 ∧N ···∧N n = p p s s 1 ∧N 2 ∧N 3 ∧N ···∧N n and similar arguments.

Proposition 1.10.7. Let (Si)i∈I be a ﬁnite family of N - sets and T another N - set. Then we have an isomorphism of N - sets: T S = (T S ) . ∧N · i ∼ · ∧N i i[∈I i[∈I Proof. The bijection is obviously given by t (s; i) (t s; i) with t T and s S , i I. ∧N ↔ ∧N ∈ ∈ i ∈

Deﬁnition 1.10.8. Let (Si)i∈I be a family of N - sets and T another N - set. A pointed map ψ : S T is called an N - multi map if i∈I i → Q ψ(...,si−1,a + si,si+1,... ) = a + ψ(...,si−1,si,si+1,... ) for all a N and s S , k I. ∈ k ∈ k ∈ Remark 1.10.9. Note that ψ being an N - multi map implies the following two properties: for all a N and s S , k I, one has ∈ k ∈ k ∈

ψ(...,a + si,...,sj ,... )= ψ(...,si,...,a + sj,... ) , and if one component of (s ) equals the distinguished point, say s = p , then ψ((s ) )= since i i∈I l l i i∈I ∞

ψ((si)i∈I )= ψ(...,sl−1,pl,sl+1,...) = ψ(...,s ,p + ,s ,...) l−1 l ∞ l+1 = + ψ((s ) ) ∞ i i∈I = . ∞

Proposition 1.10.10. Let (Si)i∈I be a ﬁnite family of N - sets and T another N - set. Every N - multi map ψ : S T gives rise to a unique N - map ψ˜ : S T such that ψπ˜ = ψ, where π i∈I i → N i → denotes the canonical projecion S S . Q i∈I i → N i V Q V Proof. Let I = 1,...,n . The map ψ˜ is a well-deﬁned binoid homomorphism by Remark 1.10.9. Its { } uniqueness follows from ψπ˜ = ψ because π is surjective. Moreover, an easy computation

ψ˜(a + s )= ψ˜((a + s ) s s ) ∧N i 1 ∧N 2 ∧N ···∧N n = ψ(a + s1,s2,...,sn)

= a + ψ((si)i∈I ) = a + ψ˜( s ) ∧N i proves that ψ˜ is an N - map. 1.10 Operations 47

The preceding proposition yields the following bijection of sets:

N - maps S T N - multi maps S T , ψ ψπ . N i → −→ i → 7−→ n i^∈I o n Yi∈I o Definition 1.10.11. Let M be a binoid. If ϕ : N M is a binoid homomorphism with ϕ(a)+ x = → x + ϕ(a) for all a N and x M, then M is called an N - binoid with respect to the structure ∈ ∈ homomorphism ϕ. An N - b i n o i d h o m o m o r p h i s m is a binoid homomorphism ψ : M M ′ → of N - binoids such that ψϕ = ϕ′, where ϕ and ϕ′ are the structure homomorphisms of M and M ′, respectively. The set of all N - binoid homomorphisms M M ′ will be denoted by hom (M,M ′). → N N - binoids together with N - binoid homomorphisms form a category BN . Let M be a commutative N - binoid via ϕ : N M. We say that M is finitely generated → as N - binoid by x ,...,x if every element f M can be written as f = ϕ(a)+ r n x for some 1 r ∈ i=1 i i a N and n N, i 1,...,r . ∈ i ∈ ∈{ } P Every N - binoid M can be considered as an N -set in a natural way with respect to the operation given by + : N M M, (a, x) a + x := ϕ(a)+ x . × −→ 7−→ Thus, an N - binoid is a binoid that is an N -set such that the operation of N is compatible with the addition of the binoid. If is a congruence on an N - binoid M, then M/ is again an N - binoid ∼ ∼ with respect to the structure homomorphism πϕ : N M M/ . → → ∼ Remark 1.10.12. A commutative N - binoid that is finitely generated as N - set is also finitely generated as N - binoid, but the converse need not be true. For instance, N∞ is finitely generated (by the element 1) as 0, - binoid but not as 0, - set. In general, a binoid M is finitely gener- { ∞} { ∞} ated as 0, - binoid if and only if it is finitely generated over every binoid N that admits a binoid { ∞} homomorphism ϕ : N M. Equivalently, M is a finitely generated binoid. → Every commuative binoid M that is finitely generated as N - binoid by x1,...,xr gives rise to a canonical binoid epimorphism

r N (Nr)∞ M, a (n ,...,n ) ϕ(a)+ n x , ∧ −→ ∧ 1 r 7−→ i i i=1 X where ϕ : N M denotes the structure homomorphism. →

Corollary 1.10.13. Let (Mi)i∈I be a family of (positive) N - binoids. The product, the direct sum, the smash product, and the bipointed union of a family of (positive) N - binoids are N - binoids.

Proof. This is an immediate consequence of Lemma 1.10.4 and the preceding observation.

Lemma 1.10.14. Let M,M ′, and L be N - binoids. Every N - binoid homomorphism ϕ : M M ′ → induces a canonical map of sets

hom (M ′,L) hom (M,L) , ψ ψϕ . N −→ N 7−→ Proof. This is clear from the diagram

N ④ ❈❈ ④④ ❈❈ ④④ ❈❈ ④④ ❈❈ }④ ϕ ψ ! M / M ′ / L. 48 1 Basic concepts of binoids

Corollary 1.10.15. Given a ﬁnite family (Mi)i∈I of N - binoids, the equivalence relation ∧ on ∼ N i∈I Mi is a congruence. In particular, the smash product of (Mi)i∈I over N is again an N - binoid with identity element 0 0 =: 0 and absorbing element =: . The V ∧N ···∧N ∧N ∞ ∧N ···∧N ∞ ∞∧N canonical inclusions

ι : M M , x 0 0 x 0 0 , k k −→ N i 7−→ ∧N ···∧N ∧N ∧N ∧N ···∧N i^∈I where x is the kth component, k I, are N - binoid homomorphisms. ∈ Proof. This is easily veriﬁed.

Every binoid can be considered as a 0, - binoid since 0, is an initial object in the category { ∞} { ∞} of binoids B. In this spirit, the smash product of a family of binoids from Section 1.8 is precisely the smash product over 0, . In particular, one cannot expect better or more results for the smash { ∞} product over an arbitrary binoid N than for the smash product (over 0, ) as listed in Lemma { ∞} 1.8.8. In fact, all these results are difficult to generalize to arbitrary N - binoids because the elements of a class x depend on the structure homomorphisms ϕ : N M , i I, see for instance Lemma ∧N i i → i ∈ 1.10.17 and Example 1.10.18 below. More precisely, t = x if t = ϕ (a )+ x with ∧N i ∧N i i j∈J ± i j i a N, j J I, where the summands ϕ (a ) are distributed in a way such that they vanish j ∈ ∈ ⊆ ± i i P after a suitable not necessarily unique shifting. Nevertheless, generators of the smash product yield generators of the smash product over N via the canonical projection M M . i∈I i → N i Example 1.10.16. Let I be finite. If (M ) is a family of N - binoidsV with generatingV sets A M , i i∈I i ⊆ i i I, then M is generated by ∈ N i V ae := 0 0 a 0 0 , i,N ∧N ···∧N ∧N ∧N ∧N ···∧N where a A is the ith entry, i I. In particular, the smash product N∞ of the family (N∞) ∈ i b ∈ N i∈I is generated by V e := 0 0 1 0 0 , i,N ∧N ···∧N ∧N ∧N ∧N ···∧N where 1 is the ith entry, i I. ∈b Lemma 1.10.17. If M is an N - binoid, then N M = M. ∧N ∼ Proof. The natural operation N M M with (a, x) ϕ(a)+ x, where ϕ : N M denotes the × → 7→ → structure homomorphism of M, induces an N - binoid homomorphism N M M with a x ∧N → ∧N 7→ ϕ(a)+ x. The inverse is obviously the N - binoid homomorphism M N M, x 0 x. → ∧N 7→ ∧N The preceding result also shows that the smash product of N - binoids may have nice properties which not all components need to fulfill. An easy example is given by

∞ ∞ ∞ Z N∞ N = Z , ∧ ∼ where all a N∞ b = are units though both components need not be so. ∧ 6 ∞∧ ∞ ∞ ∞ ∞ ∞ Example 1.10.18. Consider the N - binoid N N∞ N , where N is considered as an N - binoid ∧ once (say on the left) via ϕ :1 k and once via ϕ :1 l for some k,l 1, which means k 7→ l 7→ ≥

(a + k) N∞ b = a N∞ (b + l) ∧ ∧ ∞ ∞ ∞ ∞ for all a,b N . In particular, nk N∞ 0=0 N∞ nl for n 1. For instance, N N∞ N ∼= N if ∈ ∞ ∞∧ ∧ ∞ ≥ ∧ k =0 or l = 0. Clearly, N N∞ N is a positive N - binoid. We claim that it is also cancellative. ∧ ∞ ∞ ′ ′ For this note that each element a N∞ b N N∞ N can be written uniquely as a N∞ b , where ∧ ∈ ∧ ∧ 1.10 Operations 49 b′ is determined by b = b′ + ql with b′ < l, q N, and a′ = a + qk. In virtue of Lemma 1.4.24, it ∈ ∞ ∞ suﬃces to show that the generators 1 N∞ 0 and 0 N∞ 1 are cancellative elements in N N∞ N . So ∧ ∧ ∧ suppose that

(a N∞ b)+(1 N∞ 0) = (c N∞ d)+(1 N∞ 0) = ∞ ∧ ∧ ∧ ∧ 6 ∞∧N ∞ ∞ in N N∞ N , where a,b,c,d N. By the observation above, we may assume that b,d < l. Now ∧ ∈ the equation (a + 1) N∞ b = (c + 1) N∞ d implies a +1 = c + 1 and b = d, which shows that ∧ ∧ a N∞ b = c N∞ d. The cancellativity of 0 N∞ 1 follows similarly. In particular, 1 N∞ 0 and 0 N∞ 1 ∧ ∧ ∞∧ ∞ ∧ ∧ is the unique minimal generating set of N N∞ N by Proposition 1.4.27. ∞ ∧ ∞ In Section 1.12, we will show that N N∞ N is torsion-free if and only if k and l are coprime, cf. ∧ Lemma 1.12.16.

Definition 1.10.19. Let (Mi)i∈I be a finite family of N - binoids and L another N - binoid. An N - multi map ψ : M L that is also a binoid homomorphism will be called an N - m u l t i i∈I i → binoid homomorphism. The set of all N - multi binoid homomorphisms will be denoted by Q multN ( i∈I Mi,L). Q Let (Mi)i∈I be a finite family of N - binoids and L another N - binoid with structure homomorphisms denoted by (ϕ ) and ϕ, respectively. An N - multi binoid homomorphism ψ : M L is a i i∈I i∈I i → binoid homomorphism that satisfies Q

ψ(...,xi−1, ϕi(a)+ xi, xi+1,... )= ϕ(a)+ ψ(...,xi−1, xi, xi+1,... ) for all a N and x M , j I. ∈ j ∈ j ∈

Corollary 1.10.20. Let (Mi)i∈I be a ﬁnite family of N - binoids and L another N - binoid. Every N - multi binoid homomorphism ψ : M L gives rise to a unique N - binoid homomorphism i∈I i → ψ˜ : M L with ψπ˜ = ψ, where π : M M denotes the canonical projection. N i → Q i∈I i → N i Proof.V The existence of a unique N - multiQ map ψ˜ : V M L with ψ˜( x ) = ψ((x ) ) follows N i → ∧N i i i∈I from Proposition 1.10.10. Since V ψ˜( x + y )= ψ˜(π((x + y ) )) ∧N i ∧N i i i i∈I = ψ((xi + yi)i∈I )

= ψ((xi)i∈I )+ ψ((yi)i∈I ) = ψ˜( x )+ ψ˜( y ) ∧N i ∧N i for all x , y M , the map ψ˜ is an N - binoid homomorphism. ∧N i ∧N i ∈ N i V Corollary 1.10.21. Let (Mi)i∈I be a ﬁnite family of N - binoids and L another N - binoid. Then

hom M ,L mult M ,L . N N i ∼= N i i^∈I Yi∈I as sets.

Proof. As a consequence of Corollary 1.10.20, the map hom ( M ,L) mult ( M ,L) with N N i → N i∈I i ϕ ϕπ, where π : M M , is bijective. 7→ i∈I i → N i V Q Q V The smash product over N is a universal object, namely, it is the (ﬁnite) coproduct in the category of N - binoids BN . This is contained in the following statement. 50 1 Basic concepts of binoids

Corollary 1.10.22. Let (Mi)i∈I be a ﬁnite family of N - binoids and L a commutative N - binoid. Every family ψ : M L, i I, of N - binoid homomorphisms gives rise to a unique N - binoid i i → ∈ homomorphism ψ : M L with x ψ (x ) . N i −→ ∧N i 7−→ i i i^∈I Xi∈I In particular, if M is another binoid and ϕ : M N a binoid homomorphism, then the induced → M - binoid homomorphism

M M with x x M i −→ N i ∧M i 7−→ ∧N i i^∈I i^∈I is surjective.

Proof. The map ψ˜ : M L with (x ) ψ (x ) is an N - binoid homomorphism because i∈I i → i i∈I 7→ i∈I i i all ψ , i I, are so and L is commutative. If the structure homomorphisms are given by ϕ : N L i ∈ Q P → and ϕ : N M , i I, then ψ ϕ (a)= ϕ(a) for all a N and k I, which yields i → i ∈ k k ∈ ∈

ψ˜(...,xk−1, ϕk(a)+ xk, xk+1,...)= ψk(ϕk(a)+ xk)+ ψi(xi) Xi6=k = ϕ(a)+ ψi(xi) Xi∈I = ϕ(a)+ ψ˜((xi)i∈I ) .

Hence, ψ˜ is an N - multi binoid homomorphism so that we can apply Corollary 1.10.20 to obtain the unique N - binoid homomorphism ψ : M L with ψ( x )= ψ˜((x ) )= ψ (x ). N i → ∧N i i i∈I i∈I i i The supplement is just a special case of the ﬁrst part with (M ) considered as a family of M - binoids V i i∈I P viaϕ ˜ := ϕ ϕ : M M for all i I, L = M , and ψ the natural embeddings M ֒ M such i i → i ∈ N i i i → N i that the statement follows from Lemma 1.6.3 since ∧ ∧ . V ∼ M ≤∼ N V Example 1.10.23. Consider again the binoid N∞ N∞ discussed in Example 1.10.18 and the ∧∞ binoid homomorphisms ψ : N∞ N∞ with 1 i/ gcd(k,l), i k,l . In this situation, there is a i → 7→ ∈ { } commutative diagram ϕk N∞ / N∞

ϕl ιr

ψk ∞ ιl ∞ ∞ N / N N∞ N ∧ ▼▼ ▼▼ ψ ▼▼▼ ▼▼▼ ψl & - N∞

nl+mk with ψ(n N∞ m)= . ∧ gcd(k,l)

Proposition 1.10.24. Let (Mi)i∈I be a ﬁnite family of N - binoids with structure homomorphisms (ϕi)i∈I . If L is a commutative binoid, then

L- spec M = (ψ ) L- spec M ψ ϕ = ψ ϕ ,i,j I N i ∼ i i∈I ∈ i i i j j ∈ i∈I n i∈I o ^ Y as semigroups

Proof. This is just a restatement of Corollary 1.10.22. 1.11 Direct and projective limits 51

Corollary 1.10.25. Let M and L be N - binoids, i I. Every family ψ : M L , i I, of i i ∈ i i → i ∈ N - binoid homomorphisms gives rise to a unique N - binoid homomorphism

ψ := ψ : M L with a ψ (a ) . ∧N i N i −→ N i ∧N i 7−→ ∧N i i i^∈I i^∈I

ψi ι Proof. Apply Corollary 1.10.22 to the N - binoid homomorphisms M L i L , i I. i → i → N i ∈ V

1.11 Direct and projective limits

In this section, the existence of the limit and colimit in the category of commutative N - binoids will be proved. These can be described explicitly, but due to a lack of theory so far interesting examples for so-called direct limits (limits) and projective limits (colimits) will be discussed in subsequent chapters. Furthermore, we introduce the strong projective limit for which we also give an example.

Convention. In this section, arbitrary binoids are assumed to be commutative.

Deﬁnition 1.11.1. Let (M ) be a family of N - binoids and (I, )a directed set;thatis, I is i i∈I ≥ partially ordered with respect to and for all i, j I there is a k I with k i, j. Assume that for ≥ ∈ ∈ ≥ every pair i, j I with i j, there is an N - binoid homomorphism ϕ : M M such that ∈ ≥ ij j → i (1) ϕ = id for all i I. ii Mi ∈ (2) ϕ = ϕ ϕ for all k i j. kj ki ij ≥ ≥ Then (Mi, (ϕij )i≥j )i,j∈I is called a d i r e c t e d s y s t e m of N - binoids.

Lemma 1.11.2. If (Mi, (ϕij )i≥j )i,j∈I is a directed system of N - binoids, then

D := M , I i ∼ i]∈I . where denotes the equivalence relation on the disjoint union given by ∼ (a; i) (b; j): ϕ (a)= ϕ (b) for some k i, j , ∼ ⇔ ki kj ≥ is an N - binoid with respect to the addition deﬁned by

[(a; i)] + [(b; j)] := [(ϕki(a)+ ϕkj (b); k)] , for some k i, j. Moreover, there is a canonical N - binoid homomorphism ϕ : M D , a [(a; i)], ≥ i i → I 7→ for every i I such that ϕ = ϕ ϕ for all j i; in other words, the diagram ∈ i j ji ≥

Mi ❈❈ ❈❈ϕi ❈❈ ❈❈ ! ϕji DI ⑤⑤= ⑤⑤ ⑤⑤ϕj ⑤⑤ Mj commutes. 52 1 Basic concepts of binoids

Proof. Only the transivity of is not trivial. If (a; i) (b; j) and (b; j) (c; r), there are k,l I ∼ ∼ ∼ ∈ with k i, j and l j, r such that ϕ (a)= ϕ (b) and ϕ (b)= ϕ (c). Since I is a directed set, we ≥ ≥ ki kj lj lr have an s I with s k,l, which in particular satisﬁes s i, j, r. So we obtain ∈ ≥ ≥

ϕsi(a)= ϕsk(ϕki(a)) = ϕsk(ϕkj (b)) = ϕsj (b)= ϕsl(ϕlj (b)) = ϕsl(ϕlr(c)) = ϕsr(c) .

Hence, (a; i) (c; r). To show that the addition on D is well-deﬁned, we have to verify that ∼ I (ϕ (a)+ ϕ (b); k) (ϕ (a)+ ϕ (b); l) ki kj ∼ li lj whenever k i, j and l i, j. Again, we have an s I with s k,l, which in particular satisﬁes ≥ ≥ ∈ ≥ s i, j. Therefore, ≥

ϕsk(ϕki(a)+ ϕkj (b)) = ϕsk(ϕki(a)) + ϕsk(ϕkj (b))

= ϕsi(a)+ ϕsj (b)

= ϕsl(ϕli(a)) + ϕsl(ϕlj (b))

= ϕsl(ϕli(a)+ ϕlj (b)) .

Since all ϕ , i j, are N - binoid homomorphisms, the identity elements (0; i) of the M are glued ij ≥ i together under and so are their absorbing elements ( ; i), i I. Their equivalence classes [( ; i)] ∼ ∞ ∈ ∞ and [(0; i)] serve as an absorbing and as an identity element of DI , respectively. The supplement is clear.

The N - binoid D together with the N - binoid homomorphisms ϕ : M D , i I, has the I i i → I ∈ following universal property.

Lemma 1.11.3. Let (Mi, (ϕij )i≥j )i,j∈I be a directed system of N - binoids and (DI , (ϕi)i∈I ) as in Lemma 1.11.2. Given another N - binoid M ′ and a family ψ : M M ′ of N - binoid homomorphisms i i → such that ψ = ψ ϕ for all j i, there exists a unique N - binoid homomorphism ψ : D M ′ such i j ji ≥ I → that the diagram Mi ❇❇ ψi ❇❇ ❇❇ ϕi ❇❇ ! ψ $ ′ ϕji DI / M ⑤> ; ϕj ⑤⑤ ⑤⑤ ⑤⑤ ⑤ ψj Mj commutes for all j i. ≥ Proof. It is immediately checked that the map ψ : D M ′ with ψ([(a; k)]) := ψ (a) is the unique I → k N - binoid homomorphism.

This universal property shows that DI is the colimit in the category of commutative N - binoids, which one also calls the d i r e c t or i n d u c t i v e l i m i t and denotes it by lim Mi, cf. [8, Chapter 2.6] or [5, Chapter 5.6]. −→

Remark 1.11.4. Though diﬀerently deﬁned our construction of lim Mi coincides (of course) in case N = 0, with the one given in [14, Proposition 2.1]. Note, for−→ instance, that (a; i) (ϕ (a); k) { ∞} ∼ ki for all k i because ϕ (a)= ϕ (ϕ (a)) whenever l k. ≥ li lk ki ≥ An example of a directed system of N - binoids and its direct limit is given in Example 1.13.8. 1.11 Direct and projective limits 53

Deﬁnition 1.11.5. Let (M ) be a family of N - binoids and (I, ) a partially ordered set with i i∈I ≥ respect to . Assume that for every pair i, j I with i j, there is an N - binoid homomorphisms ≥ ∈ ≥ ϕ : M M such that ji i → j (1) ϕ = id for all i I. ii Mi ∈ (2) ϕ = ϕ ϕ for all k i j. jk ji ik ≥ ≥ Then (Mi, (ϕij )i≥j )i,j∈I is called an i n v e r s e s y s t e m of N - binoids.

Lemma 1.11.6. If (Mi, (ϕji)i≥j )i,j∈I is an inverse system of N - binoids, then

P := (a ) M ϕ (a )= a for all i j I i i∈I ∈ i ji i j ≥ n i∈I o Y is an N - subbinoid of M . Moreover, for every i I there is a canonical N - binoid homomor- i∈I i ∈ phism Q ϕ : P M , (a ) a , i I −→ i j j∈I 7−→ i such that ϕ = ϕ ϕ for all j i; in other words, the diagram i ij j ≥

Mi ⑤= O ϕi ⑤ ⑤⑤ ⑤⑤ ⑤⑤ PI ϕij ❇❇ ❇❇ ❇❇ ϕj ❇❇

Mj commutes. Proof. Both assertions follow immediately from the fact that the ϕ , j i, are N - binoid homomor- ij ≥ phisms.

The binoid P together with the N - binoid homomorphisms ϕ : P M , i I, has the following I i I → i ∈ universal property.

Lemma 1.11.7. Let (Mi, (ϕji)i≥j )i,j∈I be an inverse system of N - binoids and (PI , (ϕi)i∈I ) as in Lemma 1.11.6. Given another N - binoid M ′ and a family ψ : M ′ M of N - binoid homomorphisms i → i such that ψ = ϕ ψ for all j i, there exists a unique binoid homomorphism ψ : M ′ P such that i ij j ≥ → I the diagram 3 Mi ψi ⑤⑤> O ⑤⑤ ⑤⑤ϕi ⑤⑤ ′ ψ M / PI ϕji ❆❆ ❆❆ ϕj ❆❆ ❆❆ ψj + Mj commutes for all j i. ≥ Proof. It is immediately checked that the map ψ : M ′ P with ψ(a) := (ψ (a)) is the unique → I i i∈I N - binoid homomorphism.

This universal property shows that PI is the l i m i t in the category of commutative binoids, which one also calls the p r o j e c t i v e or i n v e r s e l i m i t and denotes it by lim Mi, cf. [8, Chapter 2.6] or [5, Chapter 5.4]. ←− 54 1 Basic concepts of binoids

Example 1.11.8. Let I = 1,...,n and M := 0, . Every family (M ) of binoids yields an { } 0 { ∞} i i∈I inverse system of 0, - binoids { ∞}

× (M0,Mi), (χ , idMi , id{0,∞}) Mi i∈I with n n lim M = (M ) M × . i i + ∪ i ←− i=1 i=1 Y Y If for instance M = N∞ for all i I 0 , then lim M = (N∞ )n (0) , which is not a ﬁnitely i ∈ \{ } i ∼ ≥1 ∪{ i∈I } generated binoid for n 2. ←− ≥ More examples of inverse systems of ( 0, - ) binoids and their projective limits are given in Ex- { ∞} ample 2.2.22 and Remark 5.1.3.

Deﬁnition 1.11.9. Let (M , (ϕ ) ) be an inverse system of N - binoids. An element (a ) i ji i≥j i,j∈I i i∈I ∈ lim M is called s t r o n g l y c o m p a t i b l e if for any two entries a and a which are = there exists i i j 6 ∞ a←−k I with k i, j such that ϕ (a )= ϕ (a ) = . Denote by s - lim M the subbinoid of lim M ∈ ≤ ki i kj j 6 ∞ i i generated by all stronlgy compatible elements. We call s - lim Mi the←− strong projective←− limit of the inverse system (Mi, (ϕji)i≥j )i,j∈I . ←−

Example 1.11.10. The strongly compatible elements of the projective limit considered in Example n × n 1.11.8 above are given by the elements of i=0 Mi and those elements of i=0(Mi)+ that are of the form ( ,a ) with at most one a = . Since the addition of such elements is again an element of ∞ i i∈I i 6 ∞ Q Q this kind, we obtain

s - lim M = M × (a ) (M ) a = for at most one i I . i ∼ i ∪ i i∈I ∈ i + i 6 ∞ ∈ ←− i∈I n i∈I o Y Y

The special case of all Mi positive yields

: s - lim Mi ∼= Mi . ←− i∈I [

In Lemma 2.3.14, we are able to describe Mred of a ﬁnitely generated binoid M by means of the strong projective limit of the inverse system deﬁned by its integral quotients.

1.12 Finitely generated commutative binoids

Finitely generated binoids were introduced in Section 1.1. There, cf. Proposition 1.4.27, we could show that a finitely generated commutative binoid M always admits a unique minimal generating set, namely M 2M , if it is positive and cancellative. Moreover, properties of a finitely generated + \ + binoid usually only depend on the properties of the generators and the relations among them, which is a finite data by Rédei’s Theorem if M is commutative, cf. Corollary 1.6.10. This makes finitely generated commutative binoids comparatively easy to describe and study. They will provide us with interesting examples especially when we study their associated algebras later. For this purpose, all one- and, to a certain extent, two-generated commutative binoids with their binoid algebras will be classified in this section.

Convention. In this section, arbitrary binoids are assumed to be commutative and, if not otherwise stated, K denotes a ring. 1.12 Finitely generated commutative binoids 55

From now on, generator will always mean binoid generator. For instance, Z∞ is a two-generated binoid (group) with generators 1 and 1, none can be dropped and there is no generating set with less − • than two elements. In general, M is an r-generated integral binoid if and only if M is an r-generated monoid. Basics on finitely generated commutative monoids can be found in [30, Chapter VI], [15, 9.3], and in the monograph [48]. § Recall, cf. Example 1.6.7, that a finitely generated binoid M with generators x1,...,xr is isomorphic to (Nr)∞/ , where ∼ε ε : (Nr)∞ M with ε(e )= x , −→ i i i 1,...,r , is the canonical binoid epimorphism. Now by Rédei’s Theorem, cf. Corollary 1.6.10, ∈ { } the congruence can be generated by finitely many relations : ε(a) = ε(b), a,b (Nr)∞. Thus, ∼ε R ∈ M is given by finitely many generators and relations, which we denote by

FC(x ,...,x )/( ,..., ) or (Nr)∞/( ,..., ) . 1 r R1 Rn R1 Rn Of course, what kind of relations occur for a specific binoid depends on the chosen generating set. In general, M/( ,..., ) is the binoid that arises from the binoid M by taking the additional relations R1 Rn ,..., into account. R1 Rn The binoid algebra of a finitely generated binoid FC(x ,...,x )/( ,..., ) is K[X ,...,X ]/I, 1 r R1 Rn 1 r where I is an ideal generated by monomials and/or binomials depending on the relations ,..., . R1 Rn This observation suggests the following definition.

Deﬁnition 1.12.1. With the notation from above, a relation on an r-generated binoid M of the form : ε(x)= , x FC , is called a m o n o m i a l r e l a t i o n . A b i n o m i a l r e l a t i o n is a relation of R ∞ ∈ r the form : ε(x)= ε(y) = , x, y FC . In addition, we say is an u n m i x e d r e l a t i o n if it is R 6 ∞ ∈ r R a monomial relation ε(x)= with # supp(x) = 1, x FC , or a binomial relation ε(x)= ε(y) = ∞ ∈ r 6 ∞ such that supp(x) = supp(y) is a singleton, x, y FC . Otherwise, is a m i x e d r e l a t i o n . ∈ r R The relation ε(x) = ε(y) = 0 is a binomial relation. A relation like ε(x) = ε(y) = will be ∞ considered as two monomial relations, namely ε(x)= and ε(y)= . ∞ ∞ Remark 1.12.2. A congruence on a ﬁnitely generated binoid is an ideal congruence if and only if it is generated by monomial relations.

Loosely speaking, a mixed relation is a relation where more than one generator is involved. We sometimes indicate this by writing (s) when s is the number of generators that appear. On one- R generated binoids there are no mixed relations. Precisely, one has three diﬀerent kinds of (unmixed) relations which may appear, namely (1) : nx = 0 for some n 2, R1 ≥ (1) : nx = for some n 2, R2 ∞ ≥ (1) : rx = sx 0, for some 1 r

Lemma 1.12.3. Let M be a finitely generated binoid and y ,...,y M. The binoid homomorphism 1 r ∈ ϕ : r y M, ϕ( r n y ) = r n y , is surjective if and only if y ,...,y generate M, and i=1h ii → ∧i=1 i i i=1 i i 1 r injective if and only if there are no mixed relations among the elements y , i I. In particular, if M is V P i ∈ finitely generated with no mixed relations among its generators y ,...,y , then M = y y . 1 r ∼ h 1i∧···∧h ri Proof. The statement concerning the injectivity follows from the definition of a mixed relation. For the surjectivity consider the following composition of binoid homomorphisms

r ϕψ : (Nr)∞ y M, −→ h ii −→ i=1 ^ where r ψ := r ψ : (Nr)∞ y ∧i=1 i −→ h ii i=1 ^ with ψ : N∞ y , 1 y , cf. Corollary 1.10.25 (with N = 0, ). Thus, ϕψ = ε : (Nr)∞ M i → h ii 7→ i { ∞} → with e y , i 1,...,r , and ϕ is surjective if and only if ε is surjective. The supplement is i 7→ i ∈ { } clear.

Since the smash product realizes the tensor product for the binoid algebras, cf. Corollary 3.5.2, we have K[ x x ]= K[ x ] K[ x ]. h 1i∧···∧h ri h 1i ⊗K ···⊗K h ℓi In order to describe all ﬁnitely generated binoids that admit a generating set with no mixed relations among the generators, it suﬃces by Lemma 1.12.3 to determine all one-generated binoids up to isomorphism, what we want to do now.

Deﬁnition 1.12.4. Let M be a binoid. An element x M has a l o o p if nx = mx for some n = m. ∈ 6 Otherwise x is loopfree. In case x has a loop, there is a smallest integer s 1 such that rx = sx ≥ for unique 0 r

Recall that all boolean binoids are torsion-free, cf. Lemma 1.4.16(2). In particular, FC(x)/(2x = x) (= 0, x, ) is torsion-free. { ∞} Lemma 1.12.6. Let M be a binoid and x M. If x has a loop of length 2 or a loop of length 1 ∈ ≥ with initial pair r +1 > r 2, then x is a torsion element and, in particular, M is not torsion-free. ≥ Proof. If r

Proposition 1.12.7. Let M be a one-generated binoid with generator x. (1) M is integral if and only if it is reduced (i.e. if x is not nilpotent). (2) x is nilpotent if and only if M = N∞/(n = ) for some n 2. ∼ ∞ ≥ (3) x is a unit if and only if M = (Z/nZ)∞ for some n 2. ∼ ≥ (4) The following statements are equivalent if x is neither nilpotent nor a unit. (a) x is a cancellative element. (b) M is inﬁnite. (c) x is loopfree. If x =2x, then this is also equivalent to 6 (d) M is torsion-free. ∞ (e) M ∼= N . Proof. (1)-(3) are clear. (4) The implications (c) (b) (a) are obvious and (a) (c) follows from ⇔ ⇒ ⇒ Example 1.12.5(3). So let 2x = x. By Lemma 1.12.6, (d) implies (c). In particular, (d) implies (a) 6 and (b), which shows that M is regular by (1) and (2). The implication (d) (e) therefore follows ⇒ from Corollary 1.4.28. Condition (e) implies everything.

~~Corollary 1.12.8. Up to isomorphism there are four diﬀerent types of one-generated binoids, namely N∞ , (Z/nZ)∞ , N∞/(r = s) , and N∞/(m = ) ∞ with n,m 2 and 1 r~~

~~These one-generated binoids can be illustrated as follows (the arrows indicate addition with the generator 1): N∞:~~

~~0 ∞ 1 2~~

~~(Z/nZ)∞: 1~~

~~0 2 ∞~~

~~n − 1~~

~~N∞/(r = s):~~

~~1 r 0 r + 1 ∞~~

~~s − 1~~

~~N∞/(m = ): ∞ 0 1 2 m − 1 ∞ 58 1 Basic concepts of binoids~~

If M is a ﬁnitely generated binoid with generators x ,...,x , then every subbinoid x is isomorphic 1 r h ki to one of the above listed one-generated binoids, and if there are no mixed relations among the xi, then M ∼= x1 xr and K[M]= K[ x1 ] K K K[ xr ] by Lemma 1.12.3 and the subsequent h i∧···∧h i h i ⊗ ···⊗ 4 h i remark. For r = 2, we obtain up to isomorphism 4 + 2 = 10 possibilities for M of this unmixed type given by

1. N∞ N∞ = (N N)∞ ❀ K[X, Y ] ∧ ∼ × 2. N∞ (Z/mZ)∞ = (N Z/mZ)∞ ❀ K[X, Y ]/(Y m 1) ∧ ∼ × − 3. N∞ N∞/(l = m) ❀ K[X, Y ]/(Y l Y m) ∧ − 4. N∞ N∞/(m = ) ❀ K[X, Y ]/(Y m) ∧ ∞ 5. (Z/nZ)∞ (Z/nZ)∞ = (Z/nZ Z/mZ)∞ ❀ K[X, Y ]/(Xn 1, Y m 1) ∧ ∼ × − − 6. (Z/nZ)∞ N∞/(l = m) ❀ K[X, Y ]/(Xn 1, Y l Y m) ∧ − − 7. (Z/nZ)∞ N∞/(m = ) ❀ K[X, Y ]/(Xn 1, Y m) ∧ ∞ − 8. N∞/(k = n) N∞/(l = m) ❀ K[X, Y ]/(Xk Xn, Y l Y m) ∧ − − 9. N∞/(k = n) N∞/(m = ) ❀ K[X,Y, ]/(Xk Xn, Y m) ∧ ∞ − 10. N∞/(n = ) N∞/(l = ) ❀ K[X, Y ]/(Xn, Y m) ∞ ∧ ∞ with k,l 1 and n,m 2, and where the identiﬁcations are given by T 1∧0 X and T 0∧1 Y . ≥ ≥ ↔ ↔ Note that Z/nZ Z/mZ = Z/nmZ if n and m are coprime so that the binoid group (Z/nmZ)∞ is a × ∼ one-generated binoid, cf. also Lemma 1.12.10 below.

The mixed relations on a two-generated binoid are (2) : kx + ly = 0 for some l, k 1, R1 ≥ (2) : kx + ly = for some l, k 1, R2 ∞ ≥ (2) : kx + ly = nx + my 0, , where (k,l) = (n,m) are not both = (0, 0), and if (k,m)=(0, 0), R3 6∈ { ∞} 6 then l,n 2 and vice versa. ≥ We have the following binoid algebras over K:

FC(x, y)/( (2)) ❀ K[X, Y ]/(XkY l 1) R1 − FC(x, y)/( (2)) ❀ K[X, Y ]/(XkY l) R2 FC(x, y)/( (2)) ❀ K[X, Y ]/(XkY l XnY m) . R3 − Of course, more than one mixed relation may occur and also combinations, except the relations (2) R1 and (2) which are mutually exclusive. R2 In the following, two-generated binoids with respect to the mixed relations above are studied in detail. We start with the binoid groups.

Lemma 1.12.9. Let M be a two-generated binoid with generators x and y. (1) M is a ﬁnite binoid group if and only if kx = 0 and ly = 0 for some k,l 2. In this case, k ≥ and l can be chosen to be minimal with this condition. (2) Let M be an inﬁnite binoid group. (a) x y = 0, . h i ∩ h i { ∞} 1.12 Finitely generated commutative binoids 59

(b) There are n,m 1 minimal with respect to nx+my =0; that is, for any other pair r, s 1 ≥ ≥ with rx + sy =0, one has r = ln and s = lm for some l 1. ≥ (c) M is torsion-free if and only if gcd(n,m)=1, where n,m 1 are minimal with respect to ≥ nx + my =0 (see (b)).

Proof. (1) M is a finite binoid group if and only if x and y are finite, which is in case x and y are h i h i units, hence cancellative, equivalent to kx = 0 and ly = 0 for some k,l 2. The statement on the ≥ minimality is obvious. (2) Let M be an infinite binoid group with nx + my = 0 for some n,m 0 (a) ≥ Suppose that rx = sy for r, s 2. Then ≥ (sn + rm)x = snx + smy = s(nx + my)=0 and similarly (sn + rm)y = 0, which obviously contradicts the assumption on M being infinite. (b) First observe that kx,ky = 0 for all k 2 by (a) because if, for instance, ky = 0 for some k 2, then 6 ≥ ≥ nx + my = 0 would imply nx = sy for some s 1 by adding y sufficiently often to it. In particular, ≥ n,m 1. Let n := min n N nx + my = 0 for some m 1 and let m 1 with n x + m y = 0. ≥ 0 ≤{ ∈ | ≥ } 0 ≥ 0 0 Suppose that nx + my =0 (= n0x + m0y) for another pair n,m 1. By the choice of n , we have n n . Since x is cancellative and kx,ky =0 ≥ 0 ≥ 0 6 for all k 1, n = n is equivalent to m = m . In particular, m is unique with n x + m y = 0. ≥ 0 0 0 0 0 The assumption mn and m>m . So one finds an l 1 such that 0 0 ≥

~~n = ln0 + r and m = lm0 + s , with 0 r~~

~~0 = nx + my = l(n0x + m0y)+ rx + sy = rx + sy .~~

By the choice of n , the case 0 r 1, then ≥ d(n′x + m′y)=0= d 0 , · where n′ = n/d and m′ = m/d, but n′x + m′y = 0 by the minimality of n and m. This shows that M 6 • is not torsion-free. For the converse, we need to prove that M is a torsion-free group if gcd(n,m) = 1. So assume that z = rx + sy M • with ∈ 0 = kz = krx + ksy for some k 2. By (b), we get kr = ln and ks = lm for some l N. Since gcd(n,m) = 1, k divides l, so ≥ ∈ we can write r = n(l/k) and s = m(l/k) with l/k N. Hence, z = rx + sy = (l/k)(nx + my) = 0. ∈ Lemma 1.12.10. Let M be a two-generated binoid that admits a generating set x, y with no mixed { } relation of the form (2). R3 (1) If M is a ﬁnite binoid group, then M is isomorphic to

~~(Z/kZ Z/lZ)∞ , × where k,l 2 are minimal with respect to kx =0= ly and gcd(k,l) 2. ≥ ≥ 60 1 Basic concepts of binoids~~

~~(2) If M is an inﬁnite binoid group and n,m 1 minimal with respect to nx+my =0 (as in Lemma ≥ 1.12.9(2b)), then M is isomorphic to~~

Z∞ or (Z Z/dZ)∞ . × In the ﬁrst case, M is torsion-free with gcd(n,m)=1, and in the latter M is not torsion-free with gcd(n,m)= d 2. ≥ Proof. (1) By Lemma 1.12.9, there are k,l 2 with kx =0= ly. If k and l are minimal with this ≥ property, we obtain from Lemma 1.12.3

M = x y = (Z/kZ)∞ (Z/lZ)∞ = (Z/kZ Z/lZ)∞ . ∼ h i ∧ h i ∼ ∧ ∼ × If gcd(k,l) = 1, then M = (Z/klZ)∞ is one-generated. Hence, gcd(k,l) 2. ∼ ≥ (2) First consider the torsion-free case, which is the case when gcd(n,m) = 1 by Lemma 1.12.9. Thus, lm + kn = 1 for some k,l Z. We may assume that k < 0 and l > 0 because the other case k > 0 ∈ and l< 0 follows by symmetry. By adding nm mn =0 to k n lm = 1 as often as necessary, one − | | − − ﬁnds a k′ < 0 and an l′ > 0 with l′m + k′n = 1. With this, the homomorphism − • ϕ : M Z , x m, y n , −→ 7−→ 7−→ − is a well-deﬁned group epimorphism since

0= nx + my 0 7−→ lx + k y lm + k y =1 | | 7−→ | | l′x + k′ y l′m + k′ y = 1 | | 7−→ | | − The injectivity of ϕ follows from the fact that 0 = ϕ(ax + by)= am bn is equivalent to a = sn and − b = sm for some s N, which gives ax + by = s(nx + my) = 0. Extending ϕ by now yields a ∈ ∞ ∞ 7→ ∞ binoid isomorphism M ∼= Z . Finally, assume that M is not torsion-free. By Lemma 1.12.9(2c), gcd(n,m)= d 2 so we can write ≥ n =nd ˜ and m =md ˜ forn, ˜ m˜ 1 with gcd(˜n, m˜ ) = 1. The latter means 1 = sm˜ + rn˜ for some r, s Z ≥ ∈ with r< 0 and s> 0 or vice versa. By symmetry, we may assume that r< 0 and s> 0. The map

~~• ψ : M Z (Z/dZ) , ψ(ix + jy) = (im˜ jn,ir˜ + js mod d) , −→ × − is a well-deﬁned group isomorphism. The surjectivity holds since~~

~~ψ(sx + ( r)y)=(1, 0) and ψ(˜nx +my ˜ )=(0, 1) . − For the injectivity, assume that ψ(ix + jy)=(0, 0). Then~~

i = kn˜ , j = km˜ , and ir + js = k(˜nr +ms ˜ ) = k 0 mod d ≡ where the latter implies ix + jy = 0. Therefore, we have shown that if M is a two-generated inﬁnite binoid group that is not torsion-free, then M = (Z Z/dZ)∞ for some d 2. ∼ × ≥ Remark 1.12.11. The two-generated binoid group Z∞ is isomorphic to FC(x, y)/(x + y = 0). In general, we have FC(x ,...,x )/(x + + x = 0) = (Zr−1)∞ , 1 r 1 ··· r ∼ where the isomorphism is given by x e when i = r, and x (e + + e ) otherwise. i 7→ i 6 r 7→ − 1 ··· r 1.12 Finitely generated commutative binoids 61

~~Proposition 1.12.12. A two-generated binoid is a binoid group if and only if it is isomorphic to~~

Z∞ or (Z Z/dZ)∞ × for d 2 or ≥ (Z/kZ Z/lZ)∞ × for k,l 2 with gcd(k,l) 2. ≥ ≥ Proof. M is a two-generated binoid group if and only if M • is a two-generated (abelian) group. By the structure theorem for ﬁnitely generated abelian groups, we know that

• M = Zd Z/n Z Z/n Z ∼ × 1 ×···× r for some d 0 and n ,...,n 2. By Lemma 1.12.10, the binoid groups listed in the proposition are ≥ 1 r ≥ the only two-generated binoid groups of this kind, all other have strictly more or less generators.

~~The associated binoid algebras of the two-generated binoid groups are given by~~

K[Z∞] = K[X, Y ]/(XY 1) , ∼ − K[(Z Z/dZ)∞] = K[X, Y ]/(XdY d 1) , × ∼ − K[((Z/kZ) (Z/lZ)∞] = K[X, Y ]/(Xk 1, Y l 1) , × ∼ − − where d 2 and gcd(k,l) 2. The second result is deduced from the fact that (1, 1) and ( 1, 0) ≥ ≥ − generate the binoid (Z Z/dZ)∞, d 2, since × ≥ d(1, 1)+(d 1)( 1, 0) = (1, 0) and (1, 1)+( 1, 0) = (0, 1) . − − − Hence, K[X, Y ] K(Z Z/dZ) with X T (1,1) and Y T (−1,0) is an epimorphism of K - algebras → × 7→ 7→ with kernel generated by XdY d 1. The other identiﬁcations are obvious. − The two-generated ﬁnite binoid groups with one mixed relation of the form (2) have a very nice R3 description in the terminology of binoids.

Proposition 1.12.13. If M is a two-generated ﬁnite binoid group that admits generators with a relation of the from (2), then M is isomophic to R3 ∞ ∞ (Z/kZ) N∞ (Z/lZ) ∧ for some k,l N with gcd(k,l) 2, where (Z/kZ)∞ is a Z∞ - binoid via ϕ :1 r and (Z/lZ)∞ is ∈ ≥ r 7→ a Z∞ - binoid via ϕ :1 s for some 2 r < k and 2 s

∞ ∞ ψ : (Z/kZ) N∞ (Z/lZ) M ∧ −→ with ι ψ = ψ , j k,l , which is surjective since ψ(1 N∞ 0) = x and ψ(0 N∞ 1) = y. On the other j j ∈{ } ∧ ∧ hand, there is a 0, - binoid homomorphism { ∞} ∞ ∞ ∞ ∞ ϕ : (Z/kZ) (Z/lZ) (Z/kZ) N∞ (Z/lZ) ∧ −→ ∧ by Corollary 1.10.22, which factors through M because ϕ(r 0) = r N∞ 0=0 N∞ s = ϕ(0 s). This ∧ ∞∧ ∧ ∞ ∧ factorization and ψ are inverse to each other. Hence, M = (Z/kZ) N∞ (Z/lZ) . ∼ ∧

~~A two-generated (ﬁnite) binoid group as in Proposition 1.12.13 with kx = ly = 0, gcd(k,l) 2, and ≥ only one more generating relation of the form rx = sy with r < k and s~~

~~sy rx y x kx = ly = 0 ∞~~

~~Its associated algebra over K is given by~~

∞ ∞ k l r s K[(Z/kZ) N∞ (Z/lZ) ] = K[X, Y ]/(X 1, Y 1,X Y ) . ∧ ∼ − − − The situation, where kx = ly (possibly 0, ) with l, k 2 is the only generating relation, can ∈ { ∞} ≥ be visualized in the following way: 1.12 Finitely generated commutative binoids 63

~~x (k + 1)x kx 2kx~~

~~0 ∞ ly 2ly y (l + 1)y~~

~~Now we consider the case where a mixed relation of this type is given.~~

Lemma 1.12.14. Let M be a two-generated binoid that admits generators x and y such that kx = ly for some l, k 2. ≥ (1) x is loopfree if and only if y is so. (2) The following statements are equivalent. (a) M is integral (b) M is reduced (c) x and y are not nilpotent.

(3) There is a surjective binoid homomorphism φ : x N∞ y M, where x and y are consid- h i ∧ h i → h i h i ered as N∞ - binoids via ϕ : 1 kx and ϕ : 1 ly, respectively, which is an isomorphism if k 7→ l 7→ kx = ly is the only mixed generating relation.

~~Proof. (1) and (2) follow from easy computations. (3) By assumption, we have a commutative diagram~~

y 9 h i ❑❑ sss ❑❑ ϕl ss ❑❑ sss ι ❑❑❑ sss ❑❑ ∞ % N ❑ x N∞ y M, ❑❑ h i ∧ O h i s9 ❑❑ ss ❑ ι ss ϕ ❑❑ ss k ❑❑ ss ❑% ss x h i which gives rise to a binoid homomorphism φ : x N∞ y M by Corollary 1.10.22. φ is surjective h i ∧ h i → because x and y generate M. If kx = ly is the only mixed generating relation, then

M = (N2)∞/( ,ke = le ) , ∼ R 1 2 where is a (maybe empty) set of unmixed relations, i.e. relations that involve only one of the R generators, e1 or e2. By Lemma 1.3.10, there is a unique binoid homomorphism

~~2 ∞ ψ : (N ) x N∞ y , e x N∞ 0 , e 0 N∞ y , −→ h i ∧ h i 1 7−→ ∧ 2 7−→ ∧ which respects the deﬁning relations of M as listed above since~~

~~ψ(ke ) = (kx) N∞ 0=0 N∞ (ly)= ψ(le ) . 1 ∧ ∧ 2~~

~~Hence, there is a binoid homomorphism ψ˜ : M x N∞ y by Lemma 1.6.3. The homomorphisms → h i ∧ h i φ and ψ˜ are inverse to each other.~~

By the preceding lemma, a two-generated binoid with generating set x, y and : kx = ly, l, k 2 ∞ { } R ≥ being the only mixed generating relation, is given by the N - binoid x N∞ y described as in the h i ∧ h i lemma. In this situation, the subbinoids x and y are isomorphic to N∞, N∞/(n = ), (Z/n)∞, h i h i ∞ 64 1 Basic concepts of binoids or N∞/(n = m) with n 2 and m 1. The possible combinations are ≥ ≥ ∞ ∞ (Z/nZ) N∞ (Z/mZ) ∧ with gcd(n,m) 2 as in Proposition 1.12.13 and ≥

∞ ∞ k l N N∞ N ❀ K[X, Y ]/(X Y ) ∧ − ∞ ∞ k l r m N /(p = ) N∞ N /(q = ) ❀ K[X, Y ]/(X Y ,X , Y ) ∞ ∧ ∞ − ∞ ∞ k l s r n m N /(r = s) N∞ N /(m = n) ❀ K[X, Y ]/(X Y ,X X , Y Y ) ∧ − − − where 1 p < k and 1 q

~~Neil’s parabola.~~

∞ ∞ In the last section, cf. Example 1.10.18, we have already shown that N N∞ N is a positive ∧ and cancellative N∞ - binoid that admits a unique minimal generating set consisting of the elements 1 N∞ 0 and 0 N∞ 1. ∧ ∧ The same holds true for the latter two N∞ - binoids in the list above since they are isomorphic to ∞ ∞ (N N∞ N )/( i,i I), where the respective additional relations i do not aﬀect these properties. ∧ ∞ R ∈ ∞ R Obviously, N /(p = ) N∞ N /(q = ) is not reduced as p(1 N∞ 0) = ∧, hence not torsion-free. ∞ ∧ ∞ ∞ ∞ ∧ ∞ By Lemma 1.12.6, the binoid N /(r = s) N∞ N /(m = n) is not torsion-free if s r 2 or n m 2. ∧ − ≥ − ≥ The remaining case r = m = 1 and s = n = 2 (the boolean case) is excluded as the mixed relation ∞ ∞ would enforce x = y. As announced in the last section, we will now determine when N N∞ N is ∧ torsion-free.

~~∞ ∞ ∞ Lemma 1.12.16. Consider the N - binoid N N∞ N from above (see also Example 1.10.18). The ∧ N∞ - binoid homomorphism~~

~~∞ ∞ ∞ nl + mk ψ : N N∞ N N with n N∞ m ∧ −→ ∧ 7−→ gcd(k,l)~~

~~∞ ∞ is injective if and only if gcd(k,l)=1. In particular, N N∞ N is torsion-free (and a subbinoid of ∧ N∞) if and only if k and l are coprime.~~

Proof. Recall that ψ is the well-deﬁned N∞ - binoid homomorphism induced by the N∞ - binoid homomorphisms N∞ N∞, 1 l/ gcd(k,l), when the codomain is the N∞ - binoid via 1 k, and → 7→ 7→ 1 k/ gcd(k,l) if the codomain is the N∞ - binoid via 1 l, cf. Example 1.10.23. 7→ 7→ 1.12 Finitely generated commutative binoids 65

Suppose that gcd(k,l) = d > 1. Then k = k′d and l = l′d for some 1 k′ < k and 1 l′ < l. ∞ ∞ ≤ ≤ Therefore, we obtain in N N∞ N , ∧ ′ ′ d(k N∞ 0) = k N∞ 0=0 N∞ l = d(0 N∞ l ) ∧ ∧ ∧ ∧ ′ ′ ∞ ∞ but k N∞ 0 = 0 N∞ l . This shows that N N∞ N is not torsion-free, in particular, ψ is not ∧ 6 ∧ ∧ injective. For the converse let k and l be coprime and

ψ(n N∞ m) = nl + mk = rl + sk = ψ(r N∞ s) ∧ ∧ with, say, n r. Then 0 (n r)l = (s m)k which implies (n r)= ck and (s m)= cl for some ≥ ≤ − − − − c> 0 because gcd(k,l) = 1. So we have n = r + ck and s = m + cl, which yields

n N∞ m = (r + ck) N∞ m = r N∞ (m + cl) = r N∞ s ∧ ∧ ∧ ∧ ∞ ∞ ∞ ∞ in N N∞ N . Hence, ψ is injective if gcd(k,l) = 1, which means that N N∞ N is a subbinoid ∧ ∧ of N∞ in this case and, in particular, torsion-free.

Proposition 1.12.17. If M is a two-generated binoid with generating set x, y , then M is isomorphic ∞ ∞ ∞ ∞ { } to N N∞ N , where N is considered as an N - binoid once via 1 k and once via 1 l for ∧ 7→ 7→ some k,l 2, if and only if M is a cancellative and positive binoid with x y = 0, . ≥ h i ∩ h i 6 { ∞} ∞ ∞ Proof. By Example 1.10.18, the binoid N N∞ N described in the proposition is positive and ∧ ∞ ∞ cancellative with unique minimal generating set 1 N∞ 0 and 0 N∞ 1. Hence, if M = N N∞ N , the ∧ ∧ ∼ ∧ isormophism is given by x 1 N∞ 0 and y 0 N∞ 1 or vice versa, which implies x y = 0, ↔ ∧ ↔ ∧ ∞ h i ∩ h i 6 { ∞} since k N∞ 0=0 N∞ l if the structure homomorphism of N is given by 1 k in the left entry and ∧ ∧ 7→ by 1 l in the right entry. 7→ For the converse note that x and y is the unique minimal generating set of M because M is positive and cancellative, cf. Proposition 1.4.27. Furthermore, M is cancellative and positive if and only if x and y are cancellative nonunits, cf. Lemma 1.4.24. By assumption, there exsits a k 2 minimal with ≥ respect to kx = ly = for some l 2. Then l is unique since the subbinoid y generated by the 6 ∞ ≥ h i cancellative nonunit y is loopfree by Proposition 1.12.7. To see that l is also minimal with respect to ly x assume that nx = my with m km 2 because ∈ h i ≥ n > k and l>m, which yields a non-trivial equation 0 = (ln km)x by the cancellativity of x and × − ∞ ∞ therefore a contradiction to x M . In particular, x y = lx = ky and M N N∞ N with 6∈ h i∩h i h i h i → ∧ x 1 N∞ 0 and y 0 N∞ 1 is a well-deﬁned binoid isomorphism because there are by assumption 7→ ∧ 7→ ∧ no other generating relations on M than : kx = ly R

Now we consider the special case of (2) : kx + ly = when k = l = 1 (i.e. x + y = ). R2 ∞ ∞ Proposition 1.12.18. Let M be a ﬁnitely generated binoid with minimal generating set x i I . { i | ∈ } If x + x = for all i = j, then M is isomorphic to i j ∞ 6 m : M with M := x i I , k k h i | ∈ ki k[=1 where I1,...,Im are the equivalence classes with respect to the relation on I generated by

~~i j : x x = 0, . ∼ ⇔ h ii ∩ h j i 6 { ∞} If #I =1, then M is isomorphic to N∞, N∞/(r = s) with 1 r~~

~~If #I 2, then k ≥ M = : N∞ (n ; i) = (n ; j) = for all i, j I k ∼ i j 6 ∞ ∈ k i∈I [k . for some n 2, i I , which may be displayed in the following way: i ≥ ∈ k~~

~~(ni; i) 0 h(1; i)i ∞ = (ni + 1; i)~~

~~In particular, if x x = 0, for all i = j, then M is isomorphic to the bipointed union of h ii ∩ h j i { ∞} 6 one-generated positive binoids.~~

Proof. The property xi + xj = for all i = j implies the positivity of M. Let Mk M be as deﬁned ∞ 6 ′ ⊆ in the proposition. If y M and z M ′ , k = k , with y,z = 0, then y + z = by the construction ∈ k ∈ k 6 6 ∞ of Mk. Hence, there exists by Proposition 1.9.9 a canonical epimorphism

~~m : M M. k −→ k[=1~~

= ′ The injectivity follows from the injectivity of the embeddings M ֒ M and the fact that M M k → k ∩ k 0, for k = k′. The description for #I = 1 is clear from Corollary 1.12.8. { ∞} 6 k For the description of M when #I 2, we may treat the M separately. So assume that x x = k k ≥ k h ii∩h j i 6 0, for i, j I and M = M. Again by Proposition 1.9.9, there exists a binoid epimorphism { ∞} ∈ k k : N∞ M with (m ; i) m x . −→ i 7−→ i i i[∈I Let i = j. By assumption, there are n ,n 2 such that n x = n x = . It follows (n + 1)x = 6 i j ≥ i i j j 6 ∞ i i n x +x = and similarly (n +1)x = . In particular, n is minimal with respect to the property j j i ∞ j j ∞ i n x = and (n + 1)x = , hence it is in particular independent of j. Therefore, we get by Lemma i i 6 ∞ i i ∞ 1.6.3 a surjective factorization

: N∞ (n ; i) = (n ; j) = for all i, j I M, i j 6 ∞ ∈ k −→ i∈I [ . which is obviously injective on the components. If rx = sx in M for i = j, then r n and s n i j 6 ≥ i ≥ j by the deﬁnition of the nis, so the homomorphism is injective. The supplement is clear.

~~By the preceding proposition, those two-generated binoids whose generators x and y fulﬁll x+y = ∞ are up to isomorphism given by the following seven binoids. For n,m 1 and l, k 2: ≥ ≥~~

N∞ : N∞ ❀ K[X, Y ]/(XY ) ∪ N∞ : N∞/(k = ) ❀ K[X, Y ]/(XY,Y k) ∪ ∞ N∞ : N∞/(m = l) ❀ K[X, Y ]/(XY,Y m Y l) ∪ − N∞/(k = ) : N∞/(l = ) ❀ K[X, Y ]/(XY,Xk, Y l) ∞ ∪ ∞ N∞/(k = ) : N∞/(m = l) ❀ K[X, Y ]/(XY,Xk, Y m Y l) ∞ ∪ − N∞/(n = k) : N∞/(m = l) ❀ K[X, Y ]/(XY,Xn Xk, Y m Y l) ∪ − − (N∞ : N∞)/((k;1)=(l; 2) = ) ❀ K[X, Y ]/(XY,Xk Y l) ∪ 6 ∞ − 1.12 Finitely generated commutative binoids 67

~~We close this section with a general study of properties of two-generated binoids.~~

Proposition 1.12.19. Let M be a two-generated binoid with generating set x, y . { } (1) If M is not cancellative but contains a non-trivial cancellative element, then M is integral if and only if one generator, say x, is regular and the other generator, y, is not nilpotent. In this case, (a) can(M)∞ = x . h i (b) M is torsion-free if and only if ky is no torsion element for all k 1. ≥ (c) M is ﬁnite if and only if M × = x • and ny = my = for some n,m 1. In this h i 6 ∞ ≥ situation, x y = 0, . h i ∩ h i { ∞} (2) If M is ﬁnite with no non-trivial cancellative elements, then x and y have a loop. If the initial pairs are 1 n

~~N∞/(r = s) : N∞/(n = m) [ or nx + ry = . 6 ∞ (b) M is integral if and only if nx + ry = . 6 ∞ (c) M is torsion-free if and only if M is boolean.~~

Proof. (1) By assumption, there is a cancellative element rx + sy = 0, r, s N. We may assume that 6 ∈ r = 0, the case s = 0 follows similarly. The implication is clear because every equation rx+sy = , 6 6 ⇐ ∞ r, s N, implies sy = since x is integral, a contradiction to y not nilpotent. So let M be integral. ∈ ∞ Then every equation x + a = x + b = , a,b M, implies rx + sy + a = rx + sy + b = , and 6 ∞ ∈ 6 ∞ hence a = b by the cancellativity of rx + sy. Therefore, x is cancellative and s = 0 because M is not cancellative, cf. Lemma 1.4.24. In particular, can(M)∞ = x , which proves (a). Since M is integral, h i x is a regular element and y not nilpotent. (b) The implication is trivial. So let ky be no torsion element for all k 1 and suppose that ⇒ ≥ n(rx + sy) = n(lx + ky) for some r,s,l,k N and n 2. We may assume that r l. By the ∈ ≥ ≥ cancellativity of x, we deduce from this equation that n((r l)x+sy)= nky. Hence, (r l)x+sy = ky − − since ky is no torsion element. Adding lx gives the desired equality rx+sy = lx+ky. (c) is clear since M is finite if and only if the subbinoids x and y are finite. For the supplement note that sy = rx h i h i implies y M ×, which yields a contradiction to M being not cancellative. Hence, x y = 0, . ∈ h i ∩ h i { ∞} (2) Clearly, x and y have a loop if M is finite. If 1 n

Example 1.12.20. Let l 2. The binoid ≥ M := FC(x, y)/(y = y + lx) is obviously positive and integral, but not cancellative because y is a non-cancellative element. On the other hand, x is a cancellative element. For this note that the relation y = y+lx implies sy = sy+mlx for all s,m 1. In particular, every element f = rx + sy M with s 1 can be written uniquely as ≥ ∈ ≥ r′x + s′y with r′

~~Example 1.12.21. The binoid~~

M := FC(x, y)/(2x = , x + y = x, 2y = y) ∞ is positive, not reduced (hence not integral), and contains no non-trivial cancellative element because both generators x and y are not cancellative and M = 0, ,x,y . { ∞ } At the end of Section 1.14, we are able to describe ﬁnitely generated binoids that are regular or regular and positive in detail, cf. Proposition 1.14.6.

~~1.13 Localization~~

We want to introduce the concept of localization to binoids since this is a very powerful tool and frequently used in commutative algebra with which we will be concerned while passing to the algebra of a binoid. Due to the fact that localization itself is a problem in non-commutative algebra we make the following agreement.

~~Convention. In this section, arbitrary binoids are assumed to be commutative.~~

Definition 1.13.1. Let M be a binoid and S a submonoid of M. We define an equivalence relation on M S as follows: for all a,a′ M and s,s′ S define × ∈ ∈ (a,s) (a′,s′): c S : a + s′ + c = a′ + s + c . ∼ ⇔ ∃ ∈ The equivalence class of (a,s) will be denoted by a-s. The set of all equivalence classes

~~M := a-s a M,s S S { | ∈ ∈ } is the localization of M at S. The localization at a submonoid that is generated by a single element f M will be abreviated by M . ∈ f 1.13 Localization 69~~

~~Lemma 1.13.2. Let M be a binoid and S a submonoid of M. The addition~~

(a-s) + (a′-s′) := (a + a′)-(s + s′) defines a binoid structure on M such that ι : M M , a a-0, is a binoid homomorphism, S S −→ S 7−→ where the elements of S become units and ker ι = a M b S : a + b = intc(M), which S { ∈ | ∃ ∈ ∞} ⊆ turns MS in an M - binoid. Proof. It is immediate to check that the operation is well-defined and that the map is a homomorphism. If s S, then (s-0) + (0-s)= s-s 0-0. Hence, s-0 M ×. ∈ ∼ ∈ S × By the preceding lemma, one has M = M × . Moreover, M = M if and only if S M , and S S+M S ⊆ M = if and only if S. S {∞} ∞ ∈ Proposition 1.13.3. Given a binoid homomorphism ϕ : M N and a submonoid S M with → ⊆ ϕ(S) N ×, there is a unique M- binoid homomorphism ϕ : M N such that the diagram ⊆ S S → ϕ M / N ④= ④④ ιS ④④ ④ ϕS ④④ MS commutes. Proof. Define ϕ (a-s) := ϕ(a)+-ϕ(s) for a M and s S, which is possible since ϕ(s) N × S ∈ ∈ ∈ by assumption. To show that this is well-defined let a-s = b-t for a,b M and s,t S. Then ∈ ∈ a + t + c = b + s + c for some c S, which implies that ϕ(a)+ ϕ(t)+ ϕ(c) = ϕ(b)+ ϕ(s)+ ϕ(c). ∈ Since ϕ(S) N ×, this is equivalent to ϕ(a)+-ϕ(s) = ϕ(b)+-ϕ(t). Hence, ϕ does not depend ⊆ S on the chosen representatives. It is an M - binoid homomorphism since ϕ is a binoid homomorphism and ϕS (a-0) = ϕ(a)+-ϕ(0) = ϕ(a). The uniqueness follows since the commutativity of the diagram requires ϕ (a-0) = ϕ(a) and ϕ (0-a)=-ϕ(a) for s S. S S ∈ The universal property of the localization has the following consequences. Corollary 1.13.4. Let M and N be binoids. If S M is a submonoid, then ⊆ N- spec M = ϕ N- spec M ϕ(S) N × S ∼ { ∈ | ⊆ } as semigroups.

Proof. By Proposition 1.5.6, the canonical binoid homomorphism ιS : M MS induces a semigroup ∗ → × homomorphism ιS : N-spec MS N-spec M with ϕ ϕιS such that ϕιS (S) N . By Proposition ∗ → 7→ ⊆ 1.13.3, ιS is an isomorphism.

Corollary 1.13.5. If S is a submonoid of M, then MS ∼= MFilt(S) as M- binoids. In particular, the localizations at the submonoids S and T of M are isomorphic as M- binoids if Filt(S) = Filt(T ). Proof. Consider the canonical binoid homomorphisms ι : M M and ι : M M . Since S → S → Filt(S) S Filt(S), one has ι(S) M × . On the other hand, if t Filt(S), say t + u = s S for u M, ⊆ ⊆ Filt(S) ∈ ∈ ∈ then (t-0)+(u-s) = (t + u)-s = 0-0 with u-s M . Hence, ι (t)= t-0 M × for all t Filt(S). Thus, ι and ι satisfy the assumptions ∈ S S ∈ S ∈ S of Proposition 1.13.3. So we obtain unique M - binoid homomorphisms M M , a-s ι(a)+ S → Filt(S) 7→ -ι(s)= a-s, and M M , a-s ι (a)+-ι (s)= a-s, which are obviously inverse to each other. Filt(S) → S 7→ S S The supplement follows from the ﬁrst part. 70 1 Basic concepts of binoids

~~By the preceding result, the set of all localizations of a binoid M is given by M F (M) =: { F | ∈F } MF(M). The operation M M := M , F ◦ G F ∩G for F, G (M), turns M into a binoid ∈F F(M)~~

M := (M , ,M ,M × ) , F(M),◦ F(M) ◦ M M where M = and M × = M. M {∞} M Corollary 1.13.6. The canonical maps F M are order preserving binoid isomorphisms between ↔ F (M) and M . F ∩ F(M),◦ Proof. Only the injectivity of F M is not obvious, but this follows from Corollary 1.13.5. 7→ F Corollary 1.13.7. If M is finitely generated, then so is M for every submonoid S M. S ⊆ Proof. Let I be finite and x i I a generating set of M. By Corollary 1.13.5, we may assume that { i | ∈ } S is a filter. Therefore, S is the submonoid generated by (x ) for some subset J I. If a-s M , j j∈J ⊆ ∈ S where a = n x and s = m x with n , m N, i I, j J, then i∈I i i j∈J j j i j ∈ ∈ ∈ P P a-s = ni(xi-0) + mj (0-xj ) . Xi∈I Xj∈J

~~This shows that (xi-0)i∈I and (0-xj )j∈J generate MS.~~

Example 1.13.8. Let S be a submonoid of the binoid M and consider S′ := S/ , where f g if ∼ ∼ Filt(f) = Filt(g). By abuse of notation, we omit the brackets and write f instead of [f] for an element of S′. Deﬁne a partial order on S′ by

f g for f,g S′ : Filt(g) Filt(f) ≥ ∈ ⇔ ⊆ and denote by ϕ : M M , a-s a-s , fg g → f 7−→ the canonical M - binoid homomorphism for f g. Note that ϕ : M M (i.e. f 0) for all ≥ f0 → f ≥ f S′, and since f + g S′ for every f,g S′, we have f + g f,g. Thus, (M , ϕ ) is a directed ∈ ∈ ∈ ≥ f fg f≥g system of M - binoids and lim M = M , f f ∼ −→ f∈S′ ] . where (a-s; f) (b-t; g): ϕ (a-s) = ϕ (b-t) for some h S′ with h f,g, together with the ∼ ⇔ hf hg ∈ ≥ canonical M - binoid homomorphisms

ϕ : M lim M , a-s [(a-s; g)] , g g → f 7→ −→ g S′, is the direct limit of this system, cf. Section 1.11. Since ψ : M M with a-s a-s ∈ f f → S 7→ is an M - binoid homomorphism for every f S′ such that ψ = ψ ϕ if f g, there is a unique ∈ g f fg ≥ M - binoid homomorphism ψ : lim M M with ψϕ = ψ , g S′, cf. Lemma 1.11.3. On the other f → S g g ∈ hand, for g S one has ϕ (g) =−→ [(g-0; g)] (lim M )× because ∈ 0 ∈ f −→ [(g-0; g)] + [(0-g; g)] = [(ϕgg(g-0) + ϕgg (0-g)); g] = [(0-0; g)] = 0 .

~~Hence, S (lim M )×. By Proposition 1.13.3, we therefore have a unique M - binoid homomorphism ⊆ f −→ 1.13 Localization 71~~

~~ϕ : M lim M with ϕι = ϕ . Thus, for every g S′ there is a commutative diagram S → f S 0 ∈ −→~~

~~Mg ① ❉ ϕg ① ❉ ψg ①① ❉❉ ①① ❉❉ ①① ψ ❉❉ {① / ! lim Mf o MS . −→ ϕ From this we deduce for a-s M , ∈ g~~

~~(ϕψ)([(a-s; g)]) = ϕ(ψ(ϕg (a-s))) = ϕ(ψg(a-s)) = ϕg(a-s) = [(a-s; g)] and (ψϕ)(a-s)= ψ(ϕ(ψg(a-s)) = ψ(ϕg (a-s)) = ψg(a-s)= a-s .~~

This shows that ψ and ϕ are inverse to each other. Hence, lim Mf ∼= MS as M - binoids. −→ To describe the smash product (over M) of two localizations of M recall the binoid structure on (M) given by F ⋆ G := Filt(F + G) for F, G (M), in which M × serves as the identity element F ∈F and M as the absorbing element. Proposition 1.13.9. Let F, G (M). Then ∈F M M = M , F ∧M G F ⋆G where the localizations are M - binoids via the canonical binoid homomorphisms ι : M M and F → F ι : M M , both given by x x-0. In particular, the canonical maps F M are order preserving G → G → ↔ F binoid isomorphisms between (M) = ( (M),⋆,M ×,M) and (M , ,M, ). F ⋆ F F(M) ∧ {∞} Proof. By Corollary 1.10.22, the M - binoid homomorphisms ψ : M M with x-f x-f and F F → F ⋆G 7→ ψ : M M with y-g y-g induce an M - binoid homomorphism G G → F ⋆G 7→ ψ : M M M , (x-f) (y-g) ψ (x-f)+ ψ (y-g) = (x + y)-(f + g) , F ∧M G −→ F ⋆G ∧M 7−→ F G where x, y M, f F , and g G. On the other hand, we have by the universal property of ∈ ∈ ∈ localization, cf. Proposition 1.13.3, an M - binoid homomorphism

φ : M M M . F ⋆G −→ F ∧M G To see this consider the binoid homomorphism ϕ : M M M with → F ∧M G ϕ(x) = (x-0) (0-0) = ι (x) (0-0) = (0-0) ι (x) = (0-0) (x-0) ( ) ∧M F ∧M ∧M G ∧M ∗ for x M. We want to show that ϕ factors through φ by applying Proposition 1.13.3. For this we ∈ need to verify that ϕ(F ⋆ G) (M M )×. So take an arbitrary x F ⋆ G. Then x + u = f + g ⊆ F ∧M G ∈ for some u M, f F , and g G. Hence, ∈ ∈ ∈ ((x-0) (0-0)) + ((u-f) 0-g) = ((x + u)-f) (0-g) ∧M ∧M ∧M = ((f + g)-f) (0-g) ∧M = ((g-0) + (0-0)) (0-g) ∧M = (0-0) ((g-0) + (0-g)) ∧M = (0-0) (0-0) ∧M

~~=0∧M . 72 1 Basic concepts of binoids~~

Hence, ϕ(x) (M M )× with inverse ϕ(x) = (u-f) (0-g). The M - binoid homomorphism ∈ F ∧M G − ∧M φ is therefore given by φ(z-x)= ϕ(z) + ( ϕ(x)) . − We claim that φ and ψ are inverse to each other. First consider an element z-x M , where ∈ F ⋆G x + u = f + g with u M, f F , and g G. Then ∈ ∈ ∈ ψφ(z-x)= ψ(ϕ(z)) + ψ( ϕ(x)) − = ψ(z 0) + ψ((u-f) (0-g)) ∧M ∧M = (z-0) + (u-(f + g)) = (z-0) + (0-x) = z-x .

By making use of ( ), we finally obtain for x, y M, f F , and g G, ∗ ∈ ∈ ∈ φψ((x-f) (y-g)) = φ((x + y)-(f + g)) ∧M = ϕ(x + y) + ( ϕ(f + g)) − = ((x-0) (y-0)) + ((0-f) (0-g)) ∧M ∧M = (x-f) (y-g) , ∧M where ϕ(f + g) = (0-f) (0-g) because ϕ(f + g) = ((f + g)-0) (0-0) = (f-0) (g-0) in − ∧M ∧M ∧M M M . F ∧M G Definition 1.13.10. Let M be a binoid and N the submonoid of M that is generated by M • . The binoid MN =: diff(M) will be called the d i f f e r e n c e b i n o i d of M.

In other words, if M is integral, then diff(M)= M • is a binoid group and diff(M)= otherwise. M {∞} Proposition 1.13.11. Let S be a submonoid of M. The binoid homomorphism ι : M M is S → S injective if and only if S consists only of regular elements. In particular, every regular binoid M is embedded into a binoid group, namely diff(M). Proof. Clear, because ι is injective if and only if a + s = b + s for a,b M, s S, implies a = b. S ∈ ∈ In the regular situation, we identify a M with a-0 M . ∈ ∈ S Corollary 1.13.12. Let M be a binoid and G a binoid group. Every binoid homomorphism ϕ : M • → G with ϕ(a) = for a M gives rise to an M - binoid homomorphism ϕ˜ : diff(M) G with 6 ∞ ∈ → ϕι˜ = ϕ, where ι : M diff(M) denotes the canonical binoid homomorphism. In particular, for every → binoid homomorphism ϕ : M G there exists a unique injective factorization → ϕ M / G , G ✎✎ π ✎✎ ✎✎ N ✎✎ ✎✎ π ✎ can ✎✎ ϕ˜ ✎✎ Ncan ✎✎ ✎✎ ι ✎ ✎✎ ✎ diff(Ncan) where N := M/ ker ϕ. 1.14 Integrality 73

Proof. Since every binoid group is regular, there exists no such binoid homomorphism ϕ for a non- integral binoid M. Therefore, we only need to consider the case of M being integral; that is, when • • × diff(M)= M • . If M is integral, the assumption ϕ(a) = for a M is equivalent to ϕ(M ) G . M 6 ∞ ∈ ⊆ So we can apply Proposition 1.13.3, which provides the unique M - binoid homomorphismϕ ˜ withϕι ˜ = ϕ. This proves the first statement. By Proposition 1.7.14(2), there is a factorization ψ : N G, can → which fulfills the assumption of the first statement since the integrality of G implies that of N by Lemma 1.7.3, and hence that of N . The injectivity of the factorization diff(N ) G follows from can can → Proposition 1.13.11.

~~Lemma 1.13.13. Let M be a regular binoid. The binoid group diﬀ(M) is torsion-free if and only if M is torsion-free.~~

Proof. Any equation nx = ny for some x, y M and n 1 in M is equivalent to n(x-0) = n(y-0) ∈ ≥ in diff(M), which implies that x = y because M is cancellative and diff(M) is torsion-free. For the converse assume that n(x-s) = 0-0 in diff(M)• . Since M is cancellative, this is equivalent to nx = ns, hence x = s because M is torsion-free, and so x-s = 0-0 = 0.

~~1.14 Integrality~~

~~In this section, we introduce integral binoid homomorphisms and related objects such as the integral closure and the normalization of a (regular) binoid.~~

Definition 1.14.1. A binoid homomorphism ϕ : M N is called i n te g r a l if for every a N there → ∈ is an integer k 1 such that ka ϕ(M). In this case, we say N is integral over M (via ϕ). ≥ ∈ Lemma 1.14.2. Let M be a subbinoid of the commutative binoid N. If N is integral over M with .× respect to the inclusion ι : M ֒ N, then N × M = M → ∩ Proof. Only the inclusion is not obvious. So assume that a N × M and denote by -a N the ⊆ ∈ ∩ ∈ inverse of a. By assumption, there is a k 1 with k(-a) M. Then c := k(-a)+(k 1)a M satisfies ≥ ∈ − ∈ a + c = 0, hence a M ×. ∈ Definition 1.14.3. Let M be a subbinoid of the binoid N. The integral closure of M in N N is the binoid M := a N na M for some integer n 1 . M is i n t e g r a l l y c l o s e d in N N { ∈ | ∈ diff(M) ≥ } if M = M. If M is regular, we call M the normalization of M, which will usually be abbreviated with M. We say M is n o r m a l if M = M.

N The integral closure is a subbinoid of N because if a,b M with na,mb M for some n,m 1, N ∈ ∈ ≥ then mn(a + b) M and so a + b M . ∈ ∈ Proposition 1.14.4. Let M be a subbinoid of the binoid N such that N is integral over M with respect to the inclusion ι : M ֒ N and let L denote an arbitrary binoid. The induced semigroup → homomorphism ι∗ : L- spec N L- spec M , ψ ψι , −→ 7−→ is a semigroup embedding if L is torsion-free. If N is commutative and L boolean, then ι∗ is a semibinoid isomorphism and, in particular,

~~N L- spec M ∼= L- spec M as semibinoids. 74 1 Basic concepts of binoids~~

Proof. The induced semigroup homomorphism ι∗ comes from Proposition 1.5.6. To prove the injectivity for L torsion-free suppose that ψ, ψ′ L-spec N with ψ = ψ′ . By assumption, there is for ∈ |M |M every a N M an integer k 2 such that ka = b M. Then kψ(a)= ψ(b)= ψ′(b)= kψ′(a), and ∈ \ ≥ ∈ hence ψ(a)= ψ′(a) since L is torsion-free. Now let N be commutative and L boolean. For the surjectivity of ι∗ let ϕ L-spec M be given. ∈ Since the inclusion ι : M ֒ N is integral and injective, there is for every a N a unique element → ∈ b M such that ka = b, where we can choose the integer k 1 to be minimal with respect to the ∈ ≥ property na M, n 1. This yields a well-deﬁned mapϕ ˜ : N L withϕ ˜(a) := ϕ(b), where b M ∈ ≥ → ∈ is this unique element for a N. By deﬁnition,ϕι ˜ = ϕ. To see thatϕ ˜ is a binoid homomorphism let ∈ a,a′ N with ka = b, k′a′ = b′, and l(a + a′)= c with b,b′,c M, and k, k′,l 1 minimal so that ∈ ∈ ≥ ϕ˜(a)= ϕ(b),ϕ ˜(a′)= ϕ(b′), andϕ ˜(a + a′)= ϕ(c). Since b,b′,c, and all their multiples lie in M and L is boolean, we obtain

~~ϕ˜(a + a′)= ϕ(c)= kk′ϕ(c)= ϕ(kk′c)=ϕ ˜(kk′c)=ϕ ˜(kk′la + kk′la′) =ϕ ˜(k′lb + klb′) = ϕ(k′lb + klb′) = k′lϕ(b)+ klϕ(b′) = ϕ(b)+ ϕ(b′) =ϕ ˜(a)+ϕ ˜(a′) .~~

By Lemma 1.5.8, L-spec M and L-spec N are boolean semibinoids with absorbing elements χM × : M L and χ × : N L, respectively. By Lemma 1.14.2, χ × ι = χ × = χ × , which shows → N → N N ∩M M that ι∗ is a semibinoid isomorphism. The supplement is obvious.

Example 1.14.5. The binoid homomorphism N∞ N∞ with n kn is integral. If L = C and → 7→ k 2, the corresponding semigroup homomorphism ≥ C-spec N∞ C-spec N∞ −→ is surjective but not injective. In case L = R and k 2, the corresponding semigroup homomorphism ≥ is neither surjective nor injective. For an algebraic closed ﬁeld L, every integral binoid embedding M ֒ N yields an integral L - algebra → homomorphism L[M] ֒ L[N], and therefore a semigroup epimorphism between the L -spectra. See → the subsequent discussion of [47, Example 2.B.16].

Now we will state well-known structure theorems on ﬁnitely generated commutative monoids trans- lated to binoids, cf. [48]. Recall that an a f f i n e m o n o i d is a monoid M which is ﬁnitely generated, cancellative, and torsion-free. Equivalently, M is a submonoid of Zd for some d 0, cf. [11, Chapter ≥ 2.A].

Proposition 1.14.6. Let M be a ﬁnitely generated binoid. (1) M is regular if and only if it is isomorphic to a subbinoid of (Zs r Z/n Z)∞ for some × k=1 k s 0 and n ,...,n 2. ≥ 1 r ≥ Q (2) M is regular and torsion-free if and only if it is isomorphic to a subbinoid of (Zs)∞ for some s 0. ≥ (3) M is positive, regular, and torsion-free if and only if it is isomorphic to a subbinoid of (Nd)∞ for some d 0. ≥ (4) If M is regular and positive, then it is isomorphic to a subbinoid of (Nd r Z/n Z)∞ for × k=1 k some d 1 and n ,...,n 2. ≥ 1 r ≥ Q 1.14 Integrality 75

Proof. (1) By Proposition 1.13.11 and Corollary 1.13.7, a finitely generated binoid is regular if and only if it can be embedded into a finitely generated abelian binoid group, namely diff(M), which is given as displayed by the structure theorem for finitely generated abelian groups. (2) follows immediately from Lemma 1.13.13 and (1). (3) By (2), M is positive, regular, and torsion-free if and only if M • is a positive affine monoid which is equivalent to M • is a submonoid of Nd for some d 0 by [11, Proposition 2.17]. ≥ (4) If M is regular and positive, we may identify M with its image under the canonical embedding ι : M ֒ diff(M)= Zs T , where T is a torsion group, cf. (1). If → × F := f Zs (f,t) M for some t T , { ∈ | ∈ ∈ } then M • = F T . Clearly, F is a submonoid of Zs, that is to say, F is an affine monoid. Furthermore, × F is positive since M = (F T )∞ is so. Indeed, if x, -x F , then (x, t), (-x, t′) M for some t,t′ T . × ∈ ∈ ∈ With n := ord T this yields

~~n((x, t)+(-x, t′)) = n(0,t + t′)=(0, 0) , which is a contradiction to M positive. Thus, F is isomorphic to a submonoid of Nd for some d 0 ≥ by (3).~~

~~2 Ideal theory in commutative binoids~~

In this chapter, we develop the ideal theory of commutative binoids in a manner parallel to that of rings. The main differences are that in the ideal theory of binoids the union of (prime) ideals is again a (prime) ideal and that every binoid is local in the sense that it admits a unique maximal ideal (with respect to ), which is prime. These two characteristics of binoids account for the main deviations ⊆ from the ideal theory of rings, cf. Remark 1.4.35 and Remark 2.2.25. Introducing the (Rees) quotient by an ideal as an important tool emphasizes again the benefit of an absorbing element. There are one- to-one corrrespondences between the prime ideals, filters, and 0, - valued binoid homomorphisms { ∞} of a binoid from which the spectrum of particular binoids can easily be deduced by taking advantage of the more general descriptions of N -spec given in Chapter 1. A treatment of the spectrum as a topological space will follow in Section 4.1. We close this section with the study on minimal prime ideals. The similarities and differences between the ideal theory of binoids and the that of rings have been intensively studied in [3]. An elaboration on the ideal theory of monoids can be found in [38]. In both publications, the theory of binoids and monoids that satisfy the ascending chain condition on ideals and the theory of primary ideals and primary decompositions have been established to a great part. Convention. In this chapter, arbitrary binoids are assumed to be commutative.

~~2.1 Ideals~~

Definition 2.1.1. Let M be a binoid. An i d e a l in M is a subset M with and x+M I ⊆ ∞ ∈ I ⊆ I for all x . ∈ I Clearly, M itself is an ideal, the u n i t i d e a l , and an ideal coincides with M if and only if it contains a unit. The condition is equivalent to = . By definition, a nonempty subset of M is an ∞ ∈ I I 6 ∅ I ideal if and only if is an M - set with respect to the addition on M restricted to : I I M , (x, a) x + a . × I −→ I 7−→ Example 2.1.2. Some sets considered before turn out to be ideals. (1) The subset is an ideal in every binoid M. {∞} (2) The set of all nonunits M = M M × of a nonzero binoid M is an ideal in M since a + b M × + \ 6∈ if a or b M . ∈ + (3) The kernel of a binoid homomorphism ϕ : M N is an ideal in M because a ker ϕ and → ∈ b M implies ϕ(a + b)= ϕ(a)+ ϕ(b)= . This is a special case of Lemma 2.1.4(1) below (with ∈ ∞ = ). For instance, if (S,p) is an N - set, the set a N a + s = p for all s S N J {∞} { ∈ | ∈ } ⊆ is an ideal in N since it is the kernel of the binoid homomorphism N (map (S,S), , id, ϕ ), → p ◦ ∞ cf. Lemma 1.10.3. Also, the kernel of the canonical projection π : M M/ is an ideal → ∼ for every ideal congruence . In particular, for a nonzero binoid M the set of all non-integral ∼ elements intc(M) and the set of all nilpotent elements nil(M) are ideals in M, cf. Lemma 1.7.6. Algorithms for computing intc(M) and nil(M) (in terms of generators of an ideal, see below) for a finitely generated binoid M are given in [50, Algorithm 9 and Algorithm 12]. 78 2 Ideal theory in commutative binoids

(4) The set + = a + b a ,b for two ideals , M is an ideal in M. In particular, I J { | ∈ I ∈J} I J ⊆ for any n 0, ≥ n = + + = a + + a a I I ··· I { 1 ··· n | i ∈ I} is an ideal (note that 0 = M according to our convention that the empty sum is 0). I (5) Clearly, every ideal in a ring R is an ideal in (R, , 1, 0). The converse is false. For instance, the · set a a =0 or a 10 is an ideal in the binoid (Z, , 1, 0) but not an ideal in the ring Z. { | | |≥ } · Deﬁnition 2.1.3. We refer to M asthe maximal ideal of M because it is the largest ideal = M + 6 in M with respect to . A binoid homomorphism ϕ : N M is l o c a l if ϕ(N ) M . ⊆ → + ⊆ + Lemma 2.1.4. Let ϕ : M N be a homomorphism of binoids. → (1) If is an ideal in N, then ϕ−1( ) is an ideal in M. J J (2) If is an ideal in M, then ϕ( )+ N = a + b a ϕ( ),b N is an ideal in N. I I { | ∈ I ∈ } Proof. Both assertions are easily veriﬁed.

Definition 2.1.5. The ideal ϕ( )+ N from Lemma 2.1.4 is called the e x t e n d e d i d e a l of by ϕ. I I Corollary 2.1.6. Let M be a binoid and S a submonoid of M. (1) If is an ideal in M , then ι−1( ) is an ideal in M. I S S I (2) If is an ideal in M, then := a-s a ,s S is an ideal in M such that = M if I IS { | ∈ I ∈ } S IS S and only if S = . ∩ I 6 ∅ Proof. These are immediate consequences of Lemma 2.1.4. For (2) note that = ι ( )+ M . IS S I S To describe the structure of the set of ideals of a binoid, we need the following definition. Definition 2.1.7. A lattice L is a set that is both a join- and meet-semilattice; that is, (L, ) and ∪ (L, ) are boolean commutative semigroups, which are related by the absorption laws a (a b)= a ∩ ∪ ∩ and a (a b) = a for all a,b L. A lattice is c o m p l e t e if any subset has a least upper and ∩ ∪ ∈ a greatest lower bound with respect to the partial order on L defined by a b : a b = b (or ⊆ ⊆ ⇔ ∪ a b : a b = a). In particular, the lattice itself is bounded by a largest and a smallest element. ⊆ ⇔ ∩ Lemma 2.1.8. The union and the intersection of ideals in a binoid M is again an ideal in M; that is, the set of all ideals is a complete lattice, partially ordered by set inclusion with largest element M and smallest element . {∞} Proof. Let ( ) be a family of ideals in M. If x , then x for some l J. Hence, Ik k∈J ∈ k∈J Ik ∈ Il ∈ x + M . If x , then x for all k J. Hence, x + M for all k J, ⊆ Il ⊆ k∈J Ik ∈ k∈J Ik ∈ Ik S ∈ ⊆ Ik ∈ which means x + M . S ⊆ k∈J IkT The subset of all properT ideals in a binoid M also defines a complete lattice, partially ordered by set inclusion, because M+ is a largest element within this set. If R is a ring, the lattice of ideals of (R, , 1, 0) is the set of all unions of ideals in the ring R. For instance, the ideal a a =0 or a 10 · { | | |≥ } in the binoid (Z, , 1, 0) is the union of all ideals nZ, n 10, in the ring Z. · ≥ Remark 2.1.9. Those lattices that arise up to isomorphism as the lattice of ideals of a binoid have been characterized by D. D. Anderson and E. W. Johnson in [3, Theorem 2.4].

Example 2.1.10. Let V be an arbitrary set. The ideal lattices of the positive binoids P(V )∩ and P(V ) can be described as follows: a subset P(V ) is an ideal in P(V ) if A and B P(V ) ∪ A⊆ ∩ ∈ A ∈ implies A B . Since A B A for all B P(V ), all subsets of A need to be contained in . ∩ ∈ A ∩ ⊆ ∈ A Hence, P(V ) is an ideal in P(V ) if and only if it is subset-closed; that is, the lattice of ideals A ⊆ ∩ is given by ( (V ), , ). Similarly, P(V ) is an ideal in P(V ) if and only if is superset-closed; S ∩ ∪ A⊆ ∪ A that is, the lattice of ideals is given by ( c(V ), , ). S ∩ ∪ 2.1 Ideals 79

Note that if V is finite, every ideal in P(V ) (resp. in P(V ) ) is uniquely determined by the A ∩ ∪ maximal (resp. minimal) sets in with respect to . A ⊆ Definition 2.1.11. By Lemma 2.1.8, there exists for every set of elements V M a smallest ideal ⊆ in M containing V , which will be denoted by V or simply by V unless the context requires M h i h i clarification. We say V is the ideal generated by V and call the elements of V generators of h i V . A set V M is a minimal set of generators of an ideal M if V = and no proper h i ⊆ I ⊆ h i I subset of V generates . An ideal is finitely generated if = V for a finite subset V M. I I I h i ⊆ A p r i n c i p a l i d e a l is an ideal generated by a singleton.

Example 2.1.12. The extended ideal of M by ϕ : M N is the ideal in N that is generated I ⊆ → by im . I Example 2.1.13. Let V be a ﬁnite set. A minimal set of generators of an ideal P(V ) is given by A⊆ ∩ the maximal sets contained in with respect to set inclusion, i.e. = max . Similarly, min A A h ⊆ Ai ⊆ A is a minimal set of generators for every ideal P(V ) . In particular, (P(V ) ) = P(V ) V = A ⊆ ∪ ∩ + \{ } V v v V and (P(V ) ) = P(V ) = v v V . For the principal ideals there are h \{ } | ∈ i ∪ + \ {∅} h | ∈ i one-to-one correspondences, namely

Principal ideals of P(V ) P Principal ideals of P(V ) , { ∩} ←→ n ←→ { ∪} J = A A J J A J A = J h i { | ⊆ } ←→ ←→ { | ⊆ } h i where the ﬁrst is order preserving and the latter reversing.

For the next result recall that the product order on Nn is defined by (a ,...,a ) (b ,...,b ) ≤ 1 n ≤ 1 n if a b for all i 1,...,n . i ≤ i ∈{ } Proposition 2.1.14. Every ideal in (Nn)∞, n 0, is finitely generated. In particular, the ideals of ≥ a finitely generated binoid are finitely generated.

Proof. It is clear that every ideal in (Nn)∞ is generated by the minimal elements of with respect I I to the product order. This is a finite set by Dickson’s Lemma, cf. [48, Theorem 5.1] or [49, Lemma 8.6], which proves the first statement. Let be an ideal in an n-generated binoid M. The preimage J := ε−1( ) under the canonical binoid epimorphism ε : (Nn)∞ M is an ideal in (Nn)∞ by Lemma I J → 2.1.4(1). By the first part, is generated by finitely many elements a ,...,a . Now the surjectivity I 1 r of ε implies that is generated by ε(a ), i 1,...,r . J i ∈{ } Example 2.1.15. By (the proof of) the preceding proposition, every ideal = in (Nn)∞ is of A 6 {∞} the form m ,...,m , where m ,...,m = min , m Nn, and denotes the product order on h 1 ri { 1 r} ≤ A i ∈ ≤ Nn. In particular, the principal ideals are given by

:= k = (N N )∞ (Nn)∞ Ik h i ≥k1 ×···× ≥kn ⊆ where k = (k ,...,k ) Nn. If k,l Nn, then = . Similarly, every ideal in 1 n ∈ ∈ Ik ∩ Il I(max(ki,li))i∈I (N∞)n is of the form m ,...,m , where m ,...,m = min , m (N∞)n, and is the product h 1 ri { 1 r} ≤ A i ∈ ≤ order on Nn extended to (N∞)n by m for all m N. In particular, the principal ideals of (N∞)n ≤ ∞ ∈ are given by ˜ := k = N∞ N∞ (N∞)n Ik h i ≥k1 ×···× ≥kn ⊆ for some k = (k ,...,k ) (N∞)n, where N∞ := . Similarly, one has ˜ ˜ = ˜ 1 n ∈ ≥∞ {∞} Ik ∩ Il I(max(ki,li))i∈I for k,l (N∞)n. Note that ϕ : (N∞)n (Nn)∞ deﬁned by e e and e , i 1,...,n , is ∈ → i 7→ i i,∞ 7→ ∞ ∈{ } a binoid epimorphism with ϕ(˜ ) + (Nn)∞ = if k Nn and otherwise. Ik Ik ∈ {∞} 80 2 Ideal theory in commutative binoids

Definition 2.1.16. Let be an ideal in M. The congruence on M defined by I ∼I a b : a = b or a,b ∼I ⇔ ∈ I is called the R e e s c o n g r u e n c e of and M/ =: M/ the (Rees) quotient of M by . The I ∼I I I extended ideal of an ideal M by the canonical projection M M/ will be denoted by / . J ⊆ → I J I Remark 2.1.17. The quotient M/ may be described as the result of collapsing into a single I I element, namely , while the elements outside remain unchanged. In particular, is an ideal ∞ I ∼I congruence and every ideal congruence is of this form since the kernel of a binoid homomorphism is an ideal, cf. Example 2.1.2(3). In this spirit, M/ can be identified with the binoid (M ) I \ I ∪ {∞} with addition given by a + b , if a + b , a + b = 6∈ I ( , otherwise, ∞ for a,b (M ) . Similarly, the extended ideal of under M M/ can be identified with ∈ \ I ∪ {∞} J → I the subset ( ( )) . J\ J ∩ I ∪ {∞} Note that the congruence induced by π : M M/ coincides with the Rees congruence of ; ∼πI I → I I that is, M/ = M/ and = ker π . In general, if ϕ : M N is a binoid epimorphism, the induced ∼πI I I I → homomorphism M/ ker ϕ N fails to be an isomorphism when = . For instance, the kernel → ∼πker ϕ 6 ∼ϕ of the binoid homomorphism ϕ : N∞ N∞ N∞, (a,b) b, is given by the ideal (a, ) a N∞ × → 7→ { ∞ | ∈ } and the quotient by (N∞ N∞)/ ker ϕ = (N∞ N) ( , ) . × ∼ × ∪{ ∞ ∞ } On the other hand, we have (N∞ N∞)/ = N∞, cf. Remark 1.6.4. See also Example 1.7.2. × ∼ϕ ∼ Example 2.1.18. (1) According to Remark 2.1.17, we obtain from the observations we made in Example 2.1.2(3):

~~c Mint = M/ int (M) and Mred = M/ nil(M) .~~

(2) Similar to the theory of semilocal rings, we have a description of the Hilbert-Samuel function H( ,M): N N in terms of the quotient by a multiple of the maximal ideal − → H(n,M) = #(M/nM ) 1 + − for a positive ﬁnitely generated binoid M.

~~(3) Let (Mi)i∈I be a ﬁnite family of binoids. Then~~

M = M and : M = M = M , i ∼ i I i ∼ i J ∼ i I ∪ J i^∈I Yi∈I . i[∈I i^∈I . Yi∈I . where = (a ) a = for some i I I { i i∈I | i ∞i ∈ } and = a a =0 for at least two diﬀerent i I . J {∧i∈I i | i 6 i ∈ } Note that = ker π∧ and = ker π : , where π∧ : Mi Mi and π : : Mi I J i∈I → i∈I i∈I → : M are the canonical binoid epimorphisms,∪ cf. Remark 1.9.6. ∪ i∈I i Q V V LemmaS 2.1.19. Let = M be an ideal in M. If M is ﬁnitely generated, positive, semifree, or I 6 cancellative, then so is M/ . I 2.1 Ideals 81

Proof. Only for the latter property there is something to show. So let M be cancellative. If [a + b]= [a + c] = [ ] in M/ for some a,b,c M, then a + b = a + c in M, which implies b = c since M is 6 ∞ I ∈ cancellative. Hence, [b] = [c].

If M is reduced and M an ideal, then M/ need not be reduced anymore. A sufficient condition I ⊆ I on such that M/ is torsion-free if M is so, is given in Lemma 2.3.8. I I Lemma 2.1.20. If M is torsion-free, then M/ is torsion-free up to nilpotence for every in M. I I Proof. Any equality [na] = [nb] = [ ] in M/ means na = nb in M. Hence, a = b in M by 6 ∞ I assumption, in particular, [a] = [b] in M/ . I Corollary 2.1.21. Let V be a (finite) set. If FC(V ) is an ideal, then FC(V )/ is (finitely) I ⊆ I generated by v V v and semifree. In particular, FC(V )/ is positive, cancellative, and { ∈ | 6∈ I} I torsion-free up to nilpotence. Conversely, every (finitely generated) semifree binoid is isomorphic to FC(V )/ for some (finite) set V and an ideal FC(V ). I I ⊆ Proof. The first statement follows from Corollary 1.4.28 with Lemma 2.1.19 and Lemma 2.1.20, and its supplement from Lemma 1.4.23. For the converse take a minimal generating set of the semifree binoid M, say V M. Then M = FC(V )/( ) , where ( ) is a family of relations of the ⊆ ∼ Rj j∈J Rj j∈J form : f = g, f,g FC(V ), cf. Example 1.6.7. Since M is semifree, all relations are monomial R ∈ Rj relations (i.e. : f = for all j J). This shows that the congruence defined by ( ) is the Rj j ∞ ∈ Rj j∈J Rees congruence of the ideal = f j J . Hence, M = FC(V )/ by Proposition 1.6.2. I h j | ∈ i ∼ I Remark 2.1.22. By Corollary 2.1.21, semifree binoids are precisely those binoids that arise from free commutative binoids modulo monomial relations. Thus, semifree binoids could also be called monomial binoids since their binoid algebras are monomial algebras and every (commutative) monomial algebra can be realized as the binoid algebra of such a binoid. A special class of monomial algebras are Stanley-Reisner algebras (or face rings), which we will encounter in Section 6.5. Proposition 2.1.23. Let M and N be binoids and an ideal in M. Every binoid homomorphism I ϕ : M N with ker ϕ factors uniquely through M/ . In particular, we have an isomorphism of → I ⊆ I semigroups, N- spec M/ = ϕ N- spec M ker ϕ . I ∼ { ∈ |I⊆ } Proof. This is a special case of Proposition 1.7.5.

Proposition 2.1.24. Let and be ideals of M. Then I J (M/ ) (M/ ) = M/( ) , I ∧M J ∼ I ∪ J where M/ is an M - binoid via the canonical projection π : M M/ , , . A A → A A∈{I J} Proof. The canonical M - binoid epimorphisms ψ : M/ M/( ) witha ¯ [a], , , A A → I ∪ J 7→ A ∈ {I J} induce by Corollary 1.10.22 an M - binoid homomorphism

ψ : (M/ ) (M/ ) M/( ) witha ¯ ¯b [¯a + ¯b] , I ∧M J −→ I ∪ J ∧M 7−→ which is surjective since ψ and ψ are so. For the injectivity note thata ¯ ¯b = if and only if I J ∧M ∞ a + b , and then [¯a + ¯b]= . If a or b this is clear. Otherwise, a and b is ∈I∪J ∞ ∈ I ∈ J 6∈ I 6∈ J equivalent to the fact thata ¯ = a and ¯b = b are singletons, so we have { } { } a¯ ¯b = a b = 0 (a + b) = (a + b) 0 ∧M ∧M ∧M ∧M and this is if and only if a + b . In particular, [¯a + ¯b] = [a + b]= . Hence, =¯a ¯b = ∞ ∈I∪J ∞ ∞ 6 ∧M a b [a + b]= a + b = if a + b , which implies the injectivity. ∧M 7→ 6 ∞ 6∈ I ∪ J 82 2 Ideal theory in commutative binoids

~~Lemma 2.1.25. Let ϕ : M N be a binoid homomorphism and M an ideal. Then → I ⊆ (M/ ) N = N/(ϕ( )+ N) I ∧M ∼ I as binoids.~~

Proof. Since ker π ker(πϕ), and hence , we have by Lemma 1.6.3 a commutative diagram I ⊆ ∼πI ≤∼πϕ of binoid homomorphisms ϕ π M / N / N/(ϕ( )+ N) ❦❦5 I ❦❦❦❦ πI ❦❦❦ ❦❦❦❦ ϕ˜ ❦❦❦❦ M/ I withϕ ˜([a]) = [ϕ(a)]. In fact, the induced homomorphismϕ ˜ is an M - binoid homomorphism because all others are so. Hence,ϕ ˜ and πϕ induce by Corollary 1.10.22 a unique M - binoid homomorphism

ψ : (M/ ) N N/(ϕ( )+ N) with [a] x [ϕ(a)+ x] . I ∧M −→ I ∧M 7−→ On the other hand, the canonical M - binoid homomorphism ι : N M N (M/ ) N with → ∧M → I ∧M ι(x) = [0] x factors through N/(ϕ( )+ N) because ϕ( )+ N ker ι. In other words, there is an ∧M I I ⊆ M - binoid homomorphism

~~Φ: N/(ϕ( )+ N) (M/ ) N with [x] [0] x . I −→ I ∧M 7→ ∧M An easy computation shows that ψ and Φ are mutually inverse.~~

Remark 2.1.26. In [3], Anderson and Johnson raised the question of which binoids admit an ideal theory most similar to that of arbitrary or noetherian rings. For the latter, they considered binoids satisfying the a s c e n d i n g c h a i n c o n d i t i o n (a.c.c.) on ideals, which means that every chain of ideals in such a binoid M becomes stationary; that is, there is a k N such that I1 ⊆ I2 ⊆ I3 ··· ∈ = for all n k. Equivalently, every ideal in M is finitely generated, which is a weaker property In Ik ≥ than being noetherian as defined in this thesis, cf. Remark 1.6.6 (Anderson and Johnson call binoids that satisfy the a.c.c. on ideals noetherian). They could prove that Krull’s intersection theorem ([13, Folgerung 5.6]) and the Artin-Rees lemma ([13, Satz 5.5]) hold true for binoids that satisfy the a.c.c. on ideals, namley, for any two ideals and I of such a binoid, one has J + N = N with N := (n + ) , I I J n\≥1 cf. [3, Theorem 3.10], and there is an m 0 such that ≥ k = ( m ) + (k m) , k m , I ∩ J I ∩ J − J ∀ ≥ cf. [3, Theorem 3.11]. An application of the Artin-Rees lemma in combination with Nakayama’s lemma is Krull’s theorem, which says that in a noetherian ring R one has an = 0 for every ideal a n≥1 { } contained in the Jacobson ideal of R ([13, Folgerung 5.8]).1 This need not be true for binoids. For T instance, the maximal ideal M+ of the finitely generated binoid

~~M = FC(x)/(2x = x)~~

1 Often, this is called Krull’s intersection theorem. 2.2 Prime ideals 83 is x , and obviously M =2M =3M = so that nM = M = . Binoids that satisfy h i + + + ··· n≥0 + + 6 {∞} nM = are called separated and are the main subject of Chapter 5. n≥0 + {∞} T Nakayama’s lemma ([13, Satz 3.19]) does not translate directly to binoids either. The canonical T projection π : N∞ (Z/nZ)∞ for n 2 turns ((Z/nZ)∞, ) into a ﬁnitely generated N∞ - set, but → ≥ ∞ for every principal ideal := k = N∞ , k 1, in N∞, one has Ik h i ≥k ≥ π( ) + (Z/nZ)∞ = (Z/nZ)∞ = 0, . Ik 6 { ∞} For monoids that satisfy the a.c.c. on ideals, The Artin-Rees lemma [38, Lemma 3.3.8], Krull’s theorem [38, Satz 3.3.9], and some version of Nakayama’s lemma [38, Lemma 3.3.1] have been proved by Kobsa, where the latter two results require in addition that the monoid M is cancellative in the sense of monoid theory (i.e. M ∞ has to be cancellative in our terms). Kobsa also studied monoids that satisfy the a.c.c. on submonoids, cf. [38, Chapter 7.4]. For this see also [30, Chapter VI.7].

Lemma 2.1.27. Let ( N be an ideal. If ϕ : N M is a local binoid homomorphism, then I → ϕ( )+ M = M. I 6 Proof. If ϕ( )+ M = M, then ϕ(a)+ b = 0 for some a and b M, and so ϕ(a) M ×, which I ∈ I ∈ ∈ contradicts ϕ being local since a N . ∈I⊆ + In [2], certain versions of Nakayama’s lemma satisﬁed by S - acts are studied.

~~2.2 Prime ideals~~

~~Deﬁnition 2.2.1. An ideal ( M is called p r i m e if M is a submonoid of M. The spectrum P \P of M, denoted by spec M, is the set of all prime ideals of M.~~

Example 2.2.2. Let M = . 6 {∞} (1) The maximal ideal M is prime since M M = M × is a subgroup of M (i.e. a submonoid that + \ + is also a group). Therefore, spec M = if and only if M = . 6 ∅ 6 {∞} (2) The ideal intc(M) of all non-integral elements is prime since M intc(M) = int(M) is a submonoid \ of M, cf. Lemma 1.4.6.

~~The set of all ideals is a complete lattice with respect to and . This translates not to the full ∩ ∪ extent to the subset spec M.~~

~~Proposition 2.2.3. The union of a set of prime ideals is again a prime ideal. In particular, the spectrum of a nonzero binoid M is a (join-) semilattice with largest element M+.~~

Proof. By Lemma 2.1.8, the union of a family of prime ideals ( ) is an ideal and since the i∈I Pi Pi i∈I intersection of submonoids of M is again a submonoid, we obtain M = (M ). S \ i∈I Pi i∈I \Pi The preceding result is the most significant difference between the idealS theories ofT binoids and rings because it contradicts the prime avoidance lemma (which states that in a ring an ideal contained in a union of n ideals with at least n 2 prime lies in one of the n ideals, cf. [9, II 1.2, Proposition 2] or − § [23, Lemma 3.3]). As this lemma has many consequences in ring theory, Proposition 2.2.3 accounts for many of the differences between the theories of binoids and rings. See for instance Proposition 2.2.24 and the subsequent remark.

Proposition 2.2.4. The intersection of a ﬁnite family of diﬀerent prime ideals is prime if and only if the family admits a unique minimal element with respect to set inclusion (i.e. a prime ideal which is contained in any other prime ideal of the family). 84 2 Ideal theory in commutative binoids

Proof. The if part is trivial. For the converse let M be a binoid and ( ) a finite family of different Pi i∈I prime ideals in M that admits strictly more than one minimal element with respect to set inclusion. Say ,..., , n 2, are those minimal elements of the family. By the minimality, there are elements P1 Pn ≥ x ( ) for all i, j 1,...,n with i = j. Thus, x n = , but by ij ∈Pi \ Pi ∩Pj ∈ { } 6 i6=j ij ∈ i=1 Pi i∈I Pi construction none of the x is contained in the intersection. Hence, the ideal is not prime. ij P T i∈I Pi T T Here is an example showing that the only if part of the preceding result may fail for an infinite familiy of prime ideals. Let I be infinite and let denote the ideal e ,e ,e ... in (NI )∞, where Pi h i i+1 i+2 i the ith component of e is 1 and all others are 0, i I. It is clear that all are prime, and since i ∈ Pi the family ( ) admits no minimal element. However, = Pi ⊃ Pi+1 ⊃ Pi+2 ⊃ · · · Pi i∈I n≥1 Pn {∞} is a prime ideal in (NI )∞. There are other very useful characterizations of prime ideals which will be T used in the following without further reference.

Lemma 2.2.5. Let = M be an ideal in M. The following statements are equivalent: P 6 (1) is prime. P (2) M/ is integral. P (3) M (M). \P∈F (4) is the kernel of a binoid homomorphism M 0, . P →{ ∞} (5) a + b for some a,b M implies that a or b . ∈P ∈ ∈P ∈P Proof. (1) (2) follows from the deﬁnition. (2) (3) If M/ is integral, then M =: F is a ⇒ ⇒ P \P submonoid of M, hence 0 F and f + g F if f,g F . On the other hand, if f + g F (i.e. ∈ ∈ ∈ ∈ f + g ), then f,g because is a prime ideal. Hence, f,g F . The equivalence (3) (4) is 6∈ P 6∈ P P ∈ ⇔ just a restatement of Proposition 1.4.41 and the implications (4) (5) (1) are obvious. ⇒ ⇒ The equivalence (1) (4) of Lemma 2.2.5 together with Proposition 1.4.41 shows, in particular, ⇔ that is a prime ideal if and only if α = χ is a binoid homomorphism. P P M\P Corollary 2.2.6. The semibinoids 0, - spec M, spec M, and (M) M are isomorphic, { ∞} F ∩ \{ } namely 0, - spec M ∼ spec M ∼ (M) M , { ∞} ←→ ←→ F \{ } α ker α 7−→ (χ =) α [ M M\P P ←− P 7−→ \P M F [ F \ ←− where the latter correspondence is inclusion reversing.

Proof. The assignments on the left-hand side are well-deﬁned and bijective by Lemma 2.2.5. The homomorphism property for both directions can be shown similarly as in Corollary 1.5.10. This also yields the statement for the assignments on the right-hand side.

~~The isomorphism 0, -spec = spec is due to Schwarz [52], see also [15, Lemma 5.54]. { ∞} ∼~~

Example 2.2.7. The ideal lattices of the binoids P(V )∩ and P(V )∪ have been described in Example 2.1.10 for arbitrary V and in Example 2.1.13 for V finite. Now we determine the subsemilattices of prime ideals for these binoids if V is finite. From the description of the filters in P(V )∩ and P(V )∪, cf. Example 1.4.38, we obtain the following description of the prime ideals and spectra:

spec P(V ) = = J V and spec P(V ) = J ( V , ∩ {PJ,∩ | ∅ 6 ⊆ } ∪ {PJ,∪ | } 2.2 Prime ideals 85 where is the subset-closed subset A P(V ) J A . Thus, PJ,∩ { ∈ | 6⊆ } = P(V j )= V j j J with = PJ,∩ \{ } h \{ } | ∈ i PJ,∩ ∪PI,∩ PJ∪I,∩ j[∈J and is the superset-closed subset A P(V ) A J . Thus, PJ,∪ { ∈ | 6⊆ } = P(V ) P(J)= j j J with = . PJ,∪ \ h{ } | 6∈ i PJ,∪ ∪PI,∪ PJ∩I,∪ In particular,

spec P(V ) = (P(V ) , , V ) and specP(V ) = (P(V ) V , , ) ∩ ∼ \ {∅} ∪ ∪ ∼ \{ } ∩ ∅ as semibinoids, where the order-preserving correspondences are given by J and J. PJ,∩ ↔ PJ,∪ ↔ This also shows that #specP = #specP = 2n 1, n 0. The prime ideals of (Nn)∞ and n,∩ n,∪ − ≥ (N∞)n, n 1, will be described in Example 2.2.21. ≥ The following corollaries are consequences of the fact that 0, -spec = spec and of results we { ∞} ∼ deduced from the induced homomorphism given in Proposition 1.5.6, which reads in terms of prime ideals as follows.

Corollary 2.2.8. Let M and N be binoids. Every binoid homomorphism ϕ : M N induces a → semigroup homomorphism ϕ∗ : spec N spec M with ϕ−1( ) such that ϕ∗ is injective if ϕ is → P 7→ P surjective.

Proof. By Proposition 1.5.6 (with L = 0, ), there is semigroup homomorphism ϕ∗ : spec N { ∞} → spec M with α α ϕ, where α is the anti-indicator homomorphism of the prime ideal spec N P 7→ P P P ∈ and ker(α ϕ)= ϕ−1( ). P P Example 2.2.9. We continue with Example 2.2.7 from above. The canonical binoid isomorphism ϕ : P(V ) ∼ P(V ) with J V J = J c induces a semibinoid isomorphism ϕ∗ : specP(V ) ∪ → ∩ 7→ \ ∩ → specP(V ) with c . ∪ PJ 7→ PJ Corollary 2.2.10. If M is an ideal, then spec M/ = spec M . More precisely, I ⊆ I ∼ {P ∈ |I ⊆P} there is an order preserving semibinoid isomorphism given by

spec(M/ ) ∼ ( spec M , ,M ) , I ←→ {P ∈ |I⊆P} ∪ + π−1( ) Q 7−→ Q / [ P I ←− P where π : M M/ denotes the canonical projection. In particular, the extended ideal of spec M → I P ∈ by π is a prime ideal in M/ if and only if . I I⊆P Proof. Note that spec M is a subsemibinoid of spec M. The isomorphism as semigroups {P ∈ |I⊆P} is just a restatement of Proposition 2.1.23 (with N = 0, ) using the identiﬁcation α of { ∞} P ↔ P Corollary 2.2.6. Since (M/ ) = M / this is an isomorphism of semibinoids. I + + I

Corollary 2.2.11. spec M ∼= spec Mred as semibinoids for every binoid M. Proof. If M = , the statement is trivial, and if M = the statement follows from Corollary {∞} 6 {∞} 2.2.10 with = nil M and the obvious fact that nil M for every spec M. (The statement I ⊆ P P ∈ also follows from Corollary 1.7.7 with N = 0, ). { ∞} 86 2 Ideal theory in commutative binoids

~~Corollary 2.2.12. Given a ﬁnite family (Mi)i∈I of binoids, there is a semibinoid isomorphism~~

spec M ∼ spec M with ( ) , i ←→ i Pi ←→ Pi i∈I i^∈I Yi∈I i[∈I b where = with = M , k = i, and = . Pi ∧k∈I Ak Ak k 6 Ai Pi Proof. By Proposition 1.8.12 (with N = 0, ) and using the identiﬁcation 0, -spec = spec, b { ∞} { ∞} there is a semigroup isomorphism

ψ : spec M spec M with α (α ι ) , i −→ i Q 7−→ Q i i∈I i^∈I Yi∈I where spec M and ι : M ֒ M , k I, are the canonical embeddings. We have Q ∈ i∈I i k k → i∈I i ∈ ker(α ι )= ι−1( ) spec M , i I. Thus, ψ( ) = (ι−1( )) . Furthermore, if spec M , i I, Q i i QV ∈ i ∈ V Q i Q i∈I Pi ∈ i ∈ then α : M 0, , a α (a ) , i −→ { ∞} ∧i∈I i 7−→ Pi i i^∈I Xi∈I where α : M 0, , i I, is the preimage of ( ) under ψ with ker α = and Pi i → { ∞} ∈ Pi i∈I i∈I Pi Pi as described in the statement. Hence, every prime ideal spec M is of this form for unique Q ∈ i∈I i S spec M , i I. In particular, (M ) is the maximal ideal of M , which showsb that bψ Pi ∈ i ∈ i∈I i + V i∈I i is an isomorphism of semibinoids. S c V Corollary 2.2.13. Given a ﬁnite family (Mi)i∈I of nonzero binoids, there is a semibinoid isomorphism spec M ∼ spec M , ˜ (( ) ; J) , i ←→ i Pi ←→ Pi i∈J Yi∈I ∅6=]J⊆I iY∈J i[∈J where ˜ = ( ) with = M , k = i, and = . Pi Ak k∈I Ak k 6 Ai Pi Proof. Consider the semigroup isomorphisms

spec M spec M spec M i −→ i −→ i ∅6=]J⊆I iY∈J ∅6=]J⊆I i^∈J Yi∈I given in Proposition 1.8.14 (with N = 0, ). We use the identiﬁcation α of Corollary 2.2.6 { ∞} P ↔ P and the description of the prime ideals of the smash product, cf. Corollary 2.2.12, for the map between the disjoint unions on the left-hand side. From these results, we deduce that the semigroup structures of the disjoint unions (as decribed in Proposition 1.8.14 for αP ) admit absorbing elements, namely

(((Mi)+)i∈I ; I) and ( i∈I (Mi)+; I). This shows that the ﬁrst map is a semibinoid isomorphism with (( ) ; J) ( ; J), where = with = M for k = i and = . By the proof Pi i∈J 7→ i∈J SPi Pi ∧k∈I Ak Ak k 6 Ai Pi of Proposition 1.8.14, the semigroupc isomorphism on the right-hand side is given by (α ; J) α ππ , S P 7→ P J where b b π α M J M π M P 0, . i −→ i −→ i −→ { ∞} Yi∈I iY∈J i^∈J Hence, if = ( ; J) as above, then ker α ππ = ˜ , where the ˜ are as in the statement. P i∈J Pi P J i∈J Pi Pi Since ( (M ) ; I) ((M ) ) this is also a semibinoid isomorphism i∈I iS+ i + i∈I S b7→ S Corollary 2.2.14.c Let (Mi)i∈I be a ﬁnite family of nonzero N - binoids with structure homomorphisms ϕ : N M , i I. Then i → i ∈ spec M = ( ) spec M ϕ−1( )= ϕ−1( ),i,j I N i ∼ Pi i∈I ∈ i i Pi j Pj ∈ i∈I n i∈I o ^ Y

2.2 Prime ideals 87 as semibinoids. More precisely, every prime ideal in M is of the form ˜ , where ˜ = N i i∈I Pi Pk ∧N Ak with = M if k = i and = otherwise, and ( ) is an element of the semibinoid on the Ak k 6 Ai Pi V Pi i∈I S right-hand side.

Proof. The isomorphism as semigroups is just a restatement of Proposition 1.10.24 (with L = 0, ) { ∞} using the identiﬁcation α of Corollary 2.2.6. The supplement follows from Corollary 2.2.12 P ↔ P since N Mi is a quotient of i∈I Mi. This shows that this is a semibinoid isomophism because (M˜ ) ((M ) ) . i∈I Vi + ↔ i + i∈I V S Corollary 2.2.15. Let (Mi)i∈I be a ﬁnite family of positive binoids. Then

spec : M = spec M i ∼ · i i[∈I i[∈I as semigroups, where the pointed union is taken over (spec Mi, (Mi)+)i∈I . In particular, every prime ideal = ( : M ) in : M is of the form = , where = (M ; i) for i = j and 6 i∈I i + i∈I i P(j) i∈I Ai Ai i 6 = ( ; j), spec M (M ) for some j I. Aj P S P ∈ Sj \{ j +} ∈ U Proof. The isomorphism and the description of the prime ideals follow from Corollary 1.9.10 (with N = 0, ) using the identiﬁcation α of Corollary 2.2.6. { ∞} P ↔P Corollary 2.2.16. Given a subbinoid M of N such that N is integral over M with respect to the inclusion ι : M ֒ N, there is a semibinoid isomorphism →

~~spec N spec M. ←→ M Q 7−→ ∩Q a N ka for some k N [ { ∈ | ∈P ∈ } ←− P~~

N In particular, spec M ∼= spec M as semibinoids for every subbinoid M of N. Proof. This is just a restatement of Proposition 1.14.4 (with L = 0, ), using the identiﬁcation { ∞} α of Corollary 2.2.6. P ↔P Corollary 2.2.17. Let M be a binoid and S a submonoid of M. The canonical homomorphism ι : M M induces an order preserving semigroup isomorphism given by S → S

spec M ( spec M S = , ) . S ←→ {P ∈ | P ∩ ∅} ∪ ι−1( ) Q 7−→ S Q ι ( )+ M [ S P S ←− P Proof. This is just a restatement of Corollary 1.13.4 (with N = 0, ), using the identification { ∞} α of Corollary 2.2.6. P ↔P Definition 2.2.18. If is a prime ideal, we use the common notation M for the localization of M P P at the filter M and ι for the canonical homomorphism M M . \P P → P With this notation Corollary 1.13.5 now translates to the following result.

Corollary 2.2.19. Let S be a submonoid of M with S. Then M = M , where is the prime ∞ 6∈ S ∼ P P ideal M Filt(S). Moreover, the unique maximal prime ideal of M is the extended ideal ι ( )+M . \ P P P P 88 2 Ideal theory in commutative binoids

Proof. By Corollary 1.13.5, we have MS ∼= MFilt(S) = MP . Thus, we only need to verify that (M )× = M (ι ( )+ M ). For the inclusion let a-s M , a M, s . If a , then P P \ P P P ⊇ ∈ P ∈ 6∈ P 6∈ P s-a M , which is the inverse of s-a, hence (a-s) (M )×. Conversely, if a-s is a unit, there are ∈ P ∈ P elements b M and t such that 0 = (a-s) + (b-t) = (a + b)-(s + t), which is equivalent to ∈ 6∈ P a + b + c = s + t + c for some c . Since s + t + c , the element a lies not in . Hence, 6∈ P 6∈ P P a-s (ι ( )+ M ). 6∈ P P P Proposition 2.2.20. If M is a binoid with generating set V M, then every prime ideal of M is of ⊆ the form A for some subset A V . In particular, #spec M 2#V . h i ⊆ ≤ Proof. If M = , the statement is clear, so let M = . Note that every ideal of the form A , {∞} 6 {∞} h i = A ( V , is prime. On the other hand, consider spec M. If = , then M is integral ∅ 6 P ∈ P {∞} and = . So let = . Every = f can be written as f = n x, where n = 0 P h∅i P 6 {∞} ∞ 6 ∈ P x∈V x x 6 for at least one but only finitely many x V . By the prime property, we get x for at least one ∈ P ∈ P x v V n =0 , which proves the first statement. The supplement is clear. ∈{ ∈ | v 6 } Example 2.2.21. Let I = 1,...,n , n 1. { } ≥ (1) The binoid (Nn)∞ admits a minimal generating set given by e , i I, cf. Proposition 1.4.27, i ∈ and all ideals e i J , J I, are prime, which implies that # spec(Nn)∞ =2n. This shows h i | ∈ i ⊆ that the bound in Proposition 2.2.20 is sharp. ∞ n (2) The binoid (N ) is generated by the 2n elements ei, ei,∞, i I, cf. Example 1.3.2(3). Thus, ′ ∈ ′ ′ the prime ideals are of the form J′,J = ei,ej,∞ i J , j J , where J ,J Pn with J = ∞ n P h | ∈ ∈ i ⊆ 6 ∅ or J = because (N ) is not integral. Since e + e = e , we have ′ = ′ ′ . 6 ∅ i i,∞ i,∞ PJ ,J PJ ,J∪J Therefore, the different prime ideals of (N∞)n are given by

′ ′ = e ,e i J , j J , PJ ,J h i j,∞ | ∈ ∈ i where = J and J ′ J. Hence, #spec(N∞)n = n 2k n . For instance, the semilattice ∅ 6 ⊆Pn ⊆ k=1 k spec(N∞)2 consisting of 8 prime ideals can be illustrated as follows P e ,e h 1 2i ⊃ ⊂ e ,e e ,e h 1 2,∞i h 2 1∞i ⊃

~~⊃ ⊂ ⊂ e e ,e e h 1i h 1,∞ 2,∞i h 2i ⊃~~

⊃ ⊂ ⊂ e e h 1,∞i h 2,∞i (3) The bipointed union : N∞ is minimally generated by (1; i), i I, cf. Proposition 1.4.27. If i∈I ∈ is a prime ideal with (1; i) , then (1; j) for every j = i because (1; i) + (1; j)= P S 6∈ P ∈P 6 ∞∈P when j = i. Hence, the prime ideals of : N∞ are given by 6 i∈I :=S(1; j) j I, j = i , Pi h | ∈ 6 i i I, and ( : N∞) = (1; i) i I . In particular, # spec : N∞ = n + 1. ∈ i∈I + h | ∈ i i∈I (4) Consider the binoid M := FC(x ,...,x )/(x + x = x ) ). If = J I, then S 1 n i i+1 i i∈I\{Sn} ∅ 6 ⊆ := x i J = x (= x ,...,x ) PJ h i | ∈ i h si h 1 si 2.2 Prime ideals 89

~~with s := max≤J. Thus, spec M is a totally ordered complete lattice given by the following sequence of principal ideals~~

x x x = M . {∞} ⊂ h 1i ⊂ h 2i⊂···⊂h ni + (5) The prime ideal intc(M) is generated by those generators of M that are not integral. (6) The spectra of the four diﬀerent types of one-generated binoids

N∞ , (Z/nZ)∞ , N∞/(r = s) , and N∞/(n = ) , ∞ cf. Corollary 1.12.8, are , 1 , , , 1 , and 1 . {{∞} h i} {{∞}} {{∞} h i} {h i} Example 2.2.22. The spectrum spec M =: I of a (ﬁnitely generated) binoid M is a partially ordered (ﬁnite) set such that (M/ , (ϕ ) ) , P PQ Q⊆P P,Q∈I where ϕ : M/ M/ PQ Q −→ P is the canonical projection for , is an inverse system of binoids. The projective limit of this Q⊆P system is given by

lim M/ = (a ) M/ ϕ (a )= a for all P P P∈I ∈ P PQ Q P Q⊆P P∈←−I n P∈I o Y together with the canonical projections ϕ : lim M/ M/ with (a ) a , I, cf. Q P → Q P P∈I 7−→ Q Q ∈ Section 1.11 (here, the partial order on I is given←− by if ). Note that if M is positive, ≥ Q≥P Q⊆P then M/M = 0, and ϕ : M/ 0, is the binoid homomorphism sending everything + { ∞} M+P P→{ ∞} but 0 to (i.e. ϕ = χ ). ∞ M+P {0} If, for instance, I contains one minimal element with respect to , say , then there is for every ⊆ Q I a binoid homomorphism ϕ : M/ M/ , which is the identity on a M/ a = P ∈ PQ Q → P { ∈ Q | 6∈ P} (M/ ) ker ϕ . Hence, Q \ PQ lim M/ = (ϕ (a)) M/ a M P PQ P∈I ∈ P ∈ P∈←−I n P∈I o Y = (a, ϕ (a)) a M/ ,a =0 (0,..., 0) = M/ . ∼ PQ P6=Q ∈ Q 6 ∪{ } ∼ Q n o : n ∞ As another example consider the binoid M = i=1 N from 2.2.21(3) (with the notation given there). ∞ ∞ The inverse system of M is given by the n + 1 binoids M/M+ = 0, , M/ i = (N ; i) ∼= N , S {∞ ∞} P i 1,...,n , together with the binoid homomorphisms ϕ : (N ; i) 0, , ϕ = id N∞ ∈{ } M+Pi →{ ∞} PiPi ( ;i) for i 1,...,n , and ϕ = id . Hence, ∈{ } M+M+ {0,∞} lim M/ = ( ,a) a (N∞ )n (0,..., 0 = (N∞ )n (0,..., 0 , P { ∞ | ∈ ≥1 }∪{ } ∼ ≥1 ∪{ } P∈←−I which is not a ﬁnitely generated binoid if n 2, though M is so. We will come back to these examples ≥ later in Example 3.2.7. Remark 2.2.23. Let N and M be binoids. Every N - point ϕ N-spec M factors uniquely through ∈ ker ϕ, cf. Lemma 1.7.3, which is a prime ideal. Indeed, we can write

N-spec M = SP , P∈spec] M 90 2 Ideal theory in commutative binoids where S := ϕ N-spec M ker ϕ = = N-spec(M/ ) is a subsemigroup of N-spec M in which P { ∈ | P} ∼ P the characteristic point χ serves as an identity element. Moreover, S + S S . See also M\P P Q ⊆ P∪Q [15, Chapter 5]. If N is a binoid group, then S = N- spec diff((M/ ) ) by Corollary 1.13.12. P ∼ P can Therefore, #N-spec M = #N- spec diff((M/ ) ) P can P∈Xspec M when spec M and N- spec diff((M/ ) ), spec M, are finite. Take for instance a finitely generated P can P ∈ torsion-free binoid M. By Lemma 1.13.13, diff((M/ ) ) is a finitely generated torsion-free group, P can which implies that diff((M/ ) )= ZlP for some l N by the structure theorem of finitely generated P can P ∈ commutative groups. Hence, #F∞-spec M = (q 1)lP . See also Proposition 4.4.7. q P∈spec M − Finally, we state a result which we will need later.P

Proposition 2.2.24. Let M be a binoid and M an ideal. If N is a submonoid of M with I ⊆ N = , then is contained in a unique ideal M which is maximal with respect to N = . ∩ I ∅ I P ⊆ ∩P ∅ Moreover, is prime. P Proof. Let be the union of all ideals with M N. Then is an ideal contained in P J I⊆J ⊆ \ P M N by Proposition 2.2.3. We have to show that is prime. For this suppose that a + b \ P ∈ P with a,b M . By the maximality of among all ideals with N = , there are elements ∈ \P P ∩P ∅ x ( a ) N and y ( b ) N. Hence, x = a + m and y = b + m′ for some m,m′ N ∈ P ∪ h i ∩ ∈ P ∪ h i ∩ ∈ because N = . Then x + y = a + b + m + m′ N , which is a contradiction to N = . ∩P ∅ ∈ ∩P ∩P ∅ Remark 2.2.25. As the analogous statement for rings, Proposition 2.2.24 implies (take N = 0 ) { } that every proper ideal in a binoid M is contained in a maximal prime ideal. This result is trivial I since M+ is prime and the upper bound of the lattice of ideals in M, cf. Lemma 2.1.8. Note that Proposition 2.2.24 is stronger than the analogous statement for rings since for rings the ideal need P not be unique. When dealing with primary ideals and primary decompositions in binoids, cf. [3] for a thorough investigation, one encounters even more consequences of the fact that the union of prime ideals is again a prime ideal. For instance, similar to a result of ring theory ([4, Proposition 4.7]), one can show that, if an ideal M admits a primary decomposition, then the set I ⊆ a M a + b for some b M = intc(M/ ) { ∈ | ∈ I ∈ \I} I ∪ I is the union of the unique associated primes of , hence intc(M/ ) is again a prime ideal. This I I ∪ I shows that a prime ideal in a binoid consisting of non-integral elements (zero-divisors in terms of ring theory) need not be contained in an associated prime.

~~2.3 Minimal prime ideals~~

Definition 2.3.1. Let M be a binoid and an ideal in M. A prime ideal containing is a I P I minimal prime of if there is no prime ideal with ( . The set of all minimal prime I Q I⊆Q P ideals of is denoted by min . The minimal prime ideals of the zero ideal are called the I M I {∞} minimal prime ideals of M and the set of all of them will be denoted by min M (instead of min ). The filter M defined by a minimal prime ideal min M is called an u l tr a fi l te r . M {∞} \P P ∈ If M is integral, the zero ideal is the only minimal prime ideal of M. There are other very useful characterizations of minimal prime ideals which will be used without further references. 2.3 Minimal prime ideals 91

Lemma 2.3.2. Let M be a binoid, an ideal in M, and spec M containing . The following I P ∈ I statements are equivalent: (1) is a minimal prime of . P I (2) M is a submonoid of M which is maximal among all submonoids S M with S = . \P ⊆ ∩ I ∅ (3) For each p , there is an element a M such that a + np for some n 1. ∈P ∈ \P ∈ I ≥ In particular, if p is an element of a minimal prime ideal of M, then a+np = for some a M P ∞ ∈ \P and n 1. ≥ Proof. Set F := M . For the implication (1) (2), only the maximality of F has to be verified \P ⇒ since F is a filter with F = by definition. So suppose that S is another submonoid with S = ∩I ∅ ∩I ∅ and F ( S. By Proposition 2.2.24, there is a unique prime ideal which is maximal with respect Q to S = . In particular, F = . The uniqueness of yields ( , but this contradicts ∩Q ∅ ∩Q ∅ Q Q P the minimality of . To prove (2) (3) take an arbitrary p and define N(p) := a + ip a P ⇒ ∈ P { | ∈ F,i N M. The set N(p) is a submonoid of M with F N(p). By the maximality of F , we have ∈ }⊆ ⊆ N(p) = , hence a + np for some a S. (Note that n = 0 is needed to have a submonoid ∩ I 6 ∅ ∈ I ∈ containing F , but a for all a F ). (3) (1) Finally, suppose that (3) holds for and there is a 6∈ I ∈ ⇒ P prime ideal with ( . Choose p such that p . By assumption, there is an element Q I⊆Q P ∈P 6∈ Q a F and an n 1 such that a + np I . Hence, p or a because is prime, but neither ∈ ≥ ∈ ⊆Q ∈Q ∈Q Q is true. Thus, has to be minimal over . P I Minimal prime ideals of an ideal always exist. To show this, we need the following lemma.

~~Lemma 2.3.3. Let be an ideal in M. If N is a submonoid of M with N = , then N is I ∩ I ∅ contained in a submonoid which is maximal with respect to this property.~~

~~Proof. This is an easy consequence of Zorn’s Lemma.~~

Proposition 2.3.4. The set of minimal prime ideals of an ideal = M in a binoid is not empty. In I 6 particular, min M = and every prime ideal contains a minimal prime ideal. 6 ∅ Proof. Since the submonoid M × does not meet , there exsists a submonoid N M which is maximal I ⊆ with respect to this property by Lemma 2.3.3. Obviously, the prime ideal M N is minimal over . \ I Deﬁnition 2.3.5. The r a d i c a l of an ideal , denoted by √ , is the set of all a M such that I I ∈ na for some n N. An Ideal M with √ = is called a r a d i c a l i d e a l . ∈ I ∈ I ⊆ I I Example 2.3.6. Obviously, the unit ideal M and all prime ideals are radical ideals. The radical of an ideal is a radical ideal since √ = √ . In particular, the ideal nil(M) is a radical ideal I I I because it is the radical of the zero ideal (i.e. = nil(M)). Therefore, nil(M) is also called the p {∞} nil radical. p Lemma 2.3.7. An ideal of M is a radical ideal if and only if M/ is reduced. I I Proof. Let be a radical ideal. If [ ]= k[a] = [ka] for some a M and k 1, then ka , which I ∞ ∈ ≥ ∈ I implies that a , so [a] = [ ]. This proves that M/ is reduced. Conversely, if M/ is reduced, it ∈ I ∞ I I is enough to verify that √ since the other inclusion is obvious. So let a M with ka for I ⊆ I ∈ ∈ I some k 1. Then k[a] = [ka] = [ ] in M/ , and hence [a] = [ ] because M/ is reduced. Thus, ≥ ∞ I ∞ I a . ∈ I Lemma 2.3.8. If M is torsion-free, then so is M/ for every radical ideal of M. I I Proof. By Lemma 2.1.20, we only need to show that M/ is reduced, which is true by Lemma 2.3.7. I 92 2 Ideal theory in commutative binoids

Proposition 2.3.9. The radical of an ideal in a binoid M is the intersection of all prime ideals in I M containing , namely I √ = . I P I⊆P∈\spec M In particular, the radical of an ideal is the intersection of its minimal primes. I Proof. If = M, the statement follows since the empty intersection is M. Therefore, let = M. I I 6 For the inclusion let a √ and be a prime ideal containing . Hence, there is an n 1 with ⊆ ∈ I P I ≥ na , but this implies a by the prime property. Conversely, assume that a √ . The ∈I⊆P ∈ P 6∈ I submonoid N := na n N of M satisﬁes N = . Applying Lemma 2.2.24, we ﬁnd a prime { | ∈ } ∩ I ∅ ideal containing with N = . In particular, a , which therefore lies not in the intersection P I ∩P ∅ 6∈ P of all prime ideals containing . I Corollary 2.3.10. = nil(M) for every nonzero binoid M. P∈min(M) P Proof. This is clearT by Proposition 2.3.9 since nil(M) = is a radical ideal which is contained {∞} in every prime ideal of M. p Corollary 2.3.11. Let M = be a reduced binoid. The following statements are equivalent: 6 {∞} (1) M is cancellative. (2) M/ is cancellative for every min M. P P ∈ Proof. (1) (2) follows from Lemma 2.1.19. To show (2) (1) suppose that a + b = a + c = ⇒ ⇒ 6 ∞ but b = c for a,b,c M. Since M is reduced, we have = by Corollary 2.3.10. 6 ∈ P∈min(M) P {∞} Thus, there is a minimal prime ideal of M with a + b = a + c . This is equivalent to [a] + [b]= P T 6∈ P [a] + [c] = [ ] in M/ , and implies [b] = [c] in M/ by the cancellativity. Since is prime, the 6 ∞ P P P elements b and c are not contained in by the choice of . Hence, b = c in M. P P The condition of being reduced is necessary in the preceding corollary. As an example consider the binoid M := FC(x,y,z)/(2x = , x + y = x + z) , ∞ which is neither reduced nor cancellative, but M/ is cancellative for all min M = x , y , z . P P ∈ {h i h i h i}

Deﬁnition 2.3.12. A subdirect product ofafamily(Mk)k∈I of binoids is a subbinoid M of their product M such that π (M) = M for all k I, where π denotes the canonical projection k∈I k k k ∈ k M M . i∈I i Q→ k QCorollary 2.3.13. Every reduced binoid M is a subdirect product of (M/ ) . P P∈min M Proof. Clearly, if M is the zero binoid, then M is a subdirect product of the empty product. So let M = and π : M M/ be the canonical projection for min M. For the injectivity of 6 {∞} P → P P ∈ π = (π ) : M M/ P P∈min M −→ P P∈Ymin M assume that π(a)= π(b) for some a,b M. This means a b which is equivalent to π (a)= π (b) ∈ ∼π P P for all min M. Therefore, a = b or a,b . Now the injectivity follows from Corollary P ∈ ∈ P∈min M P 2.3.10. In particular, M = im π is a subbinoid of M/ with π (M)= M/ . ∼ T P∈min M P P P Corollary 2.3.14. Let M be a binoid, I = specQM, ϕ : M/ M/ the canonical projection PQ Q → P for , and (M/ , (ϕ ) ) the inverse system of integral quotient binoids considered in Q⊆P P PQ Q⊆P P,Q∈I Example 2.2.22. Then s - lim M/ = M . P ∼ red ←− 2.3 Minimal prime ideals 93

Proof. By Corollary 2.2.11, we may assume that M is reduced. First observe that every element in lim M/ of the form ([a] ) , a M, is strongly compatible; here, [a] denotes the image of a M P P P∈I ∈ P ∈ under←− the canonical projection M M/ . Indeed, if [a] ,[a] = , then a and a , which → P P Q 6 ∞ 6∈ P 6∈ Q implies that a spec M, and hence = ϕ ([a] )= ϕ ([a] ). In particular, there 6∈ P ∪ Q ∈ ∞ 6 P∪Q,P P P∪Q,Q Q is a well-deﬁned binoid homomorphism

~~φ : M s - lim M/ , a ([a] ) . −→ P 7−→ P P∈I P∈←−I~~

We will show that φ is bijective to prove the statement. For the surjectivity assume that (cP )P∈I • ′ • ∈ lim M/ is strongly compatible. Thus, if cP = [a]P (M/ ) and cP′ = [b]P′ (M/ ) , then P ′ ∈ P ∈ P ϕ←− ([a] )= ϕ ′ ([b] ′ ) = for some , . This equivalent to a = b because all ϕ , , QP P QP P 6 ∞ Q⊇P P QP P⊆Q are the identity map on (M/ ) ker ϕ . The injectivity follows now from Corollary 2.3.13 since P \ QP ([a] ) = ([b] ) for a,b M implies that ([a] ) = ([b] ) , where J := min M I, hence P P∈I P P∈I ∈ P P∈J P P∈J ⊆ a = b.

~~3 Basic concepts of binoid algebras~~

~~Convention. Throughout this chapter, K denotes a ring and whenever we refer to the monoid or binoid structure of a ring we mean the one deﬁned by the multiplication unless otherwise stated.~~

In this chapter, we turn to the algebras and modules associated to binoids, N - sets, and N - binoids. Before deﬁning the binoid algebra, we recall needed facts on monoid algebras that translate to binoid algebras. For instance, a necessary and suﬃcient condition for a binoid algebra being an integral domain is given. The crucial correspondence of the sets of K - points of a binoid and its algebra, on which we will focus in the next two chapters, is stated in Proposition 3.2.3. Some considerations on the connection of the ideal theories of a binoid and its algebra are made in the third section. Finally, basic notions and properties of modules and algebras associated to an N - sets and N - binoids, respectively, are assembled in the last two sections. We want to draw the attention to two observations made in this chapter. Example 3.2.7 demonstrates that projective limits need not commute with K[ ], and Example 3.4.9 showcases how easily one drops − out of the theory of monoid algebras while still remaining in the context of binoid algebras.

~~3.1 Monoid algebras~~

Deﬁnition 3.1.1. A K - algebra A is given by a not necessarily commutative ring A and a ring homomorphism ϕ : K A such that im ϕ lies in the center of A. The ring homomorphism ϕ is → called the structure homomorphism of the K - algebra A. A ring homomorphism A A′ into → another K - algebra A′ is a K - a l g e b r a h o m o m o r p h i s m if it is compatible with the structure homomorphisms of A and A′. The set Hom (A, A′) of all K - algebra homomorphisms A A′ K- alg → will be denoted by K-Spec A when A′ = K.

We will now recall the deﬁnition of monoid algebras and list the basic results about them without giving (detailed) proofs. As a reference, we cite [51], but proofs can also be found in [46] and (for commutative monoids) in [28].

Deﬁnition 3.1.2. Let M be an arbitrary monoid. The monoid algebra KM of M over K is the associative K - algebra with K - left module basis T a M. The multiplication for these basis elements ∈ is deﬁned by using the operation of M,

~~T a T b := T a+b · for a,b M, which is then extended distributively to a multiplication on KM. In case M is a group, ∈ KM is called the g r o u p a l g e b r a of M over K.~~

The ring K is a subring of KM via the ring monomorphism K ֒ KM, r rT 0. If K = 0, there → 7→ 6 is a monoid embedding M ֒ KM with a T a such that M can be considered as a submonoid of → 7→ KM. Every element f KM can be written uniquely as ∈ a f = raT aX∈M 96 3 Basic concepts of binoid algebras with r K such that r = 0 for only ﬁnitely many a M. Moreover, KM is an M - graded a ∈ a 6 ∈ K - algebra KM = KT a , aM∈M where KT a := rT a r K . The elements of the K - module KT a are called homogeneous of { | ∈ } degree a. The element 0 is of every degree by convention. Clearly, unless K = 0 the monoid algebra KM is commutative if and only if M is commutative (and K, what we always assume).

Example 3.1.3. The polynomial algebra K[X i I] = K(N(I)) over K is a monoid algebra. i | ∈ ∼ By the following proposition, the monoid algebra is a universal object and therefore unique up to isomorphism.

Proposition 3.1.4. Given a K - algebra A and a monoid homomorphism φ : M (A, ), there exists → · a unique K - algebra homomorphism φ : KM A with φ˜(T a)= φ(a) for all a M. → ∈ Proof. See [51, Korollar 52.2].

Corollary 3.1.5. Let M be a monoid. (1) Let α : K L be a homomorphism of rings and ϕ : M N a monoid homomorphism. Then → → there is a unique ring homomorphism KM LN with ra α(r)ϕ(a). → 7→ (2) If a is an ideal in K, then KM/aKM ∼= (K/a)M. (3) If S is a multiplicative system in K, then (KM)S ∼= KSM. (4) Given a ﬁnite family (Mi)i∈I of monoids, there is a canonical isomorphism

K M KM . i ∼= K i Yi∈I Oi∈I (5) If A is a K - algebra, then A KM = AM. ⊗K ∼ Proof. For a more precise demonstration see [51, 52, Beispiel 2,3 and 4], [51, 80, Beispiel 13], and § § [51, 81, Beispiel 11]. Outline: the ﬁrst assertion is an easy consequence of the universal property § (Proposition 3.1.4). The second follows from (1) with N = M, ϕ = id , and α = π : K K/a. M → Finally, one may easily derive that the three remaining isomorphisms are given by the canonical K - algebra homomorphism ( r T a)/s (r /s)T a and by the two canonical K - algebra S a∈M a 7→ a∈M a homomorphisms T (ai)i∈I a and b r T a (r b)T a. 7→ ⊗Pi∈I i ⊗K aP∈M a 7→ a∈M a The following proposition determines underP which conditionsP the monoid algebra contains no zero- divisors.

~~Proposition 3.1.6. Let M be a monoid and K = 0. The monoid algebra KM is a domain if and 6 only if K is an integral domain and M ∞ is torsion-free and cancellative.~~

~~Proof. See [30, Theorem 8.1] or [11, Theorem 4.18].~~

~~3.2 Binoid algebras~~

For the sake of completeness, we recall the deﬁnition of the binoid algebra introduced at the beginning of Section 1.1 (cf. page 10). Since we will encounter only commutative binoid algebras in the subsequent chapters, two examples of non-commutative algebras that can be realized as binoid algebras are given at the end of this section, namely matrix algebras and path algebras. 3.2 Binoid algebras 97

Deﬁnition 3.2.1. Let M be a binoid. The binoid algebra of M is deﬁned to be the quotient algebra KM/(T ∞) =: K[M] , where (T ∞) is the ideal in KM generated by the element T ∞. In general, if is an ideal in M, we I will denote the ideal (resp. K - submodule) of KM generated by T a, a , by ∈ I K := (T a a ) KM I | ∈ I ⊆ and by K[ ] := K /(T ∞) = (T a a ) K[M] I I | ∈ I ⊆ the associated ideal (resp. K - submodule) of K[M].

By definition, K[M] may be identified with the set of all formal sums r T a with A M • a∈A a ⊆ finite and r K, where the multiplication is generated by a ∈ P a+b a b rasbT , if a + b = , raT sbT = 6 ∞ · (0 , otherwise.

~~The K - module isomorphism a K[M] ∼= KT • aM∈M gives rise to a graded K - algebra homomorphism~~

KM K[M] −→ with ker = KT ∞. In this vein, the binoid algebra K[M] emerges from the monoid algebra KM by glueing together the absorbing elements of M and KM. Thus, the notion of binoid algebras generalizes that of monoid algebras. Given an ideal M, the ideals K and K[ ] are monomial ideals of KM I ⊆ I I and K[M], respectively. If K = 0, the composition M ֒ KM K[M] yields a binoid embedding 6 → → ι : M K[M] , a T a , M −→ 7−→ such that M can be considered as a subbinoid of (K[M], , 1, 0). · Example 3.2.2. (1) For the zero binoid, we obtain K[ ] = 0 and for the trivial binoid K[ 0, ]= K. {∞} • { ∞} (2) If M is an integral binoid, then K[M] ∼= KM . In particular, K[M/M+] is the group algebra KM ×, see also Corollary 3.2.8(3) bellow. (3) The binoid algebras of the one-generated binoids, cf. Corollary 1.12.8, as well as those of some two-generated binoids were described up to isomorphism in Section 1.12.

Proposition 3.2.3. Given a binoid M, a K - algebra A, and a binoid homomorphism ϕ : M A, → there is a unique K - algebra homomorphism φ : K[M] A such that the diagram → ϕ M / A ②< ②② ι ②② ② φ ②② K[M] commutes. 98 3 Basic concepts of binoid algebras

~~In particular, if A = K, then K- spec M ∼= K- Spec K[M] as semigroups.~~

Proof. Consider the composition of the canonical maps M KM π K[M]. By the universal property → → of the monoid algebra, cf. Proposition 3.1.4, ϕ induces a K -algebra homomorphismϕ ˜ : KM A → with rT a α(r)ϕ(a), where α : K A is the structure homomorphism, r K, and a M. Since 7→ → ∈ ∈ ker π = KT ∞ kerϕ ˜, the K -algebra homomorphismϕ ˜ induces a ring homomorphism φ : K[M] A ⊆ → with φπ =ϕ ˜ such that rT a α(r)ϕ(a), r K, a M, which shows that φ is a K -algebra homo- 7→ ∈ ∈ morphism.

~~The binoid algebra is uniquely determined by its universal property.~~

Corollary 3.2.4. Given a ring homomorphism α : K L and a binoid homomorphism ϕ : M M ′, → → • there is a unique ring homomorphism φ : K[M] L[M ′] with φ(rT a)= α(r)T ϕ(a), r K, a M . → ∈ ∈ Proof. By Proposition 3.2.3, ϕ induces the K - algebra homomorphism φ˜ of the following commutative diagram ϕ M / M ′

~~ιM ιM′~~

~~ φ˜ α˜ K[M] / K[M ′] / L[M ′] .~~

Then φ =α ˜φ˜ is the unique ring homomorphism, whereα ˜ : rT a α(r)T a. 7→ Remark 3.2.5. There are two special cases of Corollary 3.2.4. (1) If ϕ = id , there is a unique ring homomorphism α[M]: K[M] L[M]. It is easily veriﬁed M → that every subset of L that generates L as a K - module (or K -algebra) generates L[M] as a K[M] - module (or K[M]-algebra), and that every linear independent set of elements of L over K is linearly independent in L[M] over K[M]. In particular, bases retain unchanged while switching to binoid algebras. Moreover, if α is surjective, then so is α[M]. (2) The other specialization is when L = K . In this case there is a unique K - algebra (!) homomorphism K[ϕ]: K[M] K[N] . −→ ϕ is injective or surjective if and only if K[ϕ] is so. In particular, K[ ] is a covariant functor − from the category of binoids to the category of graded K - algebras.

֒ Example 3.2.6. Let M be a commutative binoid. By Corollary 2.3.13, there is an embedding M → P ∈min M M/P for every reduced binoid M, which induces an injective K - algebra homomorphism K[M] K M/P . Q → P ∈min M

Example 3.2.7. Q Let ((Mi)i∈I , (ϕji)i≥j )i,j∈I be an inverse system of binoids. Then (Ai, (fji)i≥j )i,j∈I with A := K[M ] and f := K[ϕ ]: A A deﬁnes an inverse system of K -algebras. The i i ji ji i → j projective limit

lim K[M ] = (F ) K[M ] f (F )= F for all i j i i i∈I ∈ i ji i j ≥ ←− n i∈I o Y might but need not coincide with K[lim Mi]. However, there is always an embedding ←− K[lim M ] lim K[M ] ,T (ai)i∈I = T ai (T ai ) , i −→ i i 7−→ i i∈I ←− ←− i∈I Y 3.2 Binoid algebras 99 coming from the family g : lim M M K[M ], j I, of binoid homomorphisms which satisfy j i → j → j ∈ f g = g for all i, j I. Hence,←− there is a unique binoid embedding ij j i ∈ ϕ : lim M lim K[M ] with (a ) (T ai ) i −→ i i i∈I 7−→ i i∈I ←− ←− that factors through K[lim Mi] by Proposition 3.2.3. As an example, consider←− the inverse system ((M/ ) , (ϕ ) ) of Example 2.2.22, where P P∈I PQ Q⊆P P,Q∈I M is a commutative binoid, I = spec M, and ϕ : M/ M/ are the canonical projections for PQ Q → P . Returning to the two particular cases discussed there, we obtain the following: Q⊆P If min M = , then lim M/ = M/ , and similarly we get {Q} P Q ←− lim K[M/ ] = (f (F )) K[M/ ] F K[M] = K[M/ ] , P PQ P∈I ∈ P ∈ ∼ Q P∈←−I n P∈I o Y where f is the K - algebra epimorphism K[ϕ ]: K[M/ ] K[M/ ] induced by the canonical PQ PQ Q → P projection ϕ : M/ M/ . In particular, K[lim M ] = lim K[M/ ]. PQ Q → P P∈I P P If M = : n N∞, then lim M/ = (N∞ )n (0←−,..., 0) , cf. Example←− 2.2.22. Therefore, i=1 P ≥1 ∪{ } ←− S K[lim M/ ] = K KXa , P ⊕ ←− ∞6=a∈(N∞ )n M≥1 where Xa = Xa1 Xan for = a = (a ,...,a ) (N∞ )n. Here the inverse system of K - algebras 1 ··· n ∞ 6 1 n ∈ ≥1 is given by A := K[M/M ] = K and A := K[N∞] = K[X], i 1,...,n , with K - algebra 0 + i ∼ ∈ { } homomorphisms f = id , i 0, 1,...,n and f : K[X] K, F const(F ), i = 0, where ii Ai ∈ { } i0 → → 6 const(F ) denotes the constant term of F K[M]. Thus, ∈ n lim K[M/ ] = (c, F ,...,F ) K K[M ] F K[X], const(F )= c K,i 1,...,n . P 1 n ∈ × i i ∈ i ∈ ∈{ } P∈←−I n i=1 o Y In particular, lim K[M/ ] = K[lim M/ ] for n 2 since for instance (0,X,..., 2X) lim K[M/ ] P 6 P ≥ ∈ P −← .[ /lies not in the←− image of K[lim M/←−] ֒ lim K[M P → P ←− ←− Corollary 3.2.8. Let M be a binoid.

~~(1) If N is a subbinoid of M, the binoid algebra K[N] is a K - subalgebra of K[M].~~

(2) If a is an ideal in K, then (K/a)[M] ∼= K[M]/aK[M]. (3) If M is commutative and an ideal in M, then K[M/ ] = K[M]/K[ ] = KM/K . I I ∼ I ∼ I (4) If M is commutative and S is a subbinoid of M, then S := T a a S deﬁnes a multiplicative −1 { | ∈ } system in K[M] and there is an isomorphism S (K[M]) ∼= K[MS]. e (5) If A is a K - algebra, then A K K[M] = A[M]. ⊗ ∼ e (6) If S is a multiplicative system in K, then K[M]S ∼= KS[M].

Proof. (1) Is clear. Let π : K K/a be the canonical surjection. The kernel of the induced → ring epimorphism K[π]: K[M] (K/a)[M], cf. Corollary 3.2.4, consists of all f K[M] with → ∈ coeﬃcients in a, which proves (2). (3) The binoid epimorphism M M/ induces a K - algebra → I a homomorphism K[M] K[M/ ] by Corollary 3.2.4. Its kernel is given by • KT = K[ ], hence → I a∈I I K[M/ ] = K[M]/K[ ]. The latter isomorphism is clear. (4) By Corollary 3.2.4, the canonical map I ∼ I L ι : M M induces a K - algebra homomorphism ˜ι : K[M] K[M ]. Let ι : K[M] S−1(K[M]) S → S S → S → denote the canonical ring homomorphism. Since ι(S) (S−1(K[M]))×, there is by the universal ⊆ e e 100 3 Basic concepts of binoid algebras property of localization (for rings) the following commutative diagram

~~ι K[M] / S−1(K[M]) , qq qqq ˜ιS qq qq ψ e xqq K[MS] where the induced K - algebra homomorphism ψ is an isomorphism with inverse given by rewriting elements of K[MS] in the following way~~

′ n n aj -0 fi-0 n aj n fi n a rj T T rj T T rj T aj -fj i6=j i6=j j=1 rj T = n n = T fi-0 7−→ T fi T f j=1 j=1 i=1Q j=1 i=1Q P X X X Q Q with a′ = a + f M and f = n f S. This is an element in S−1(K[M]). (5) We j i6=j i ∈ i=1 i ∈ have A K[M] = (A KM)/(1 T ∞) = AM/(T ∞) = A[M], where the isomorphism in the ⊗K P∼ ⊗K ⊗ P ∼ ∼ middle is due to Corollary 3.1.5(2). (6) We have K[M] = K K[M] = K [eM], where the latter S ∼ S ⊗K ∼ S isomorphism is due to (5).

Corollary 3.2.9. Let M be an ideal and e an idempotent element such that K is a I ⊆ ∈ I I K - algebra with identity e. Then ϕ : KM K K[M/ ] deﬁned by φ(x) = (ex, π(x)), where π : → I× I KM K[M/ ] denotes the canonical homomorphism, is an isomorphism of algebras. In particular, → I KM = K K[M] as K - algebras. ∼ × Proof. By assumption, there is a K - algebra isomorphism KM = K (1 e)KM, and hence ∼ I × − (1 e)KM = KM/K = K[M/ ] by Corollary 3.2.8. Since π((1 e)x) = π(x) for every x − ∼ I ∼ I − ∈ KM and ker π (1 e)KM = K (1 e)KM = 0, the restriction of π to (1 e)KM is an ∩ − I ∩ − − isomorphism (1 e)KM = K[M/ ] from which the statement follows. The case = KT ∞ proves − ∼ I I the supplement.

~~Theorem 3.2.10. The binoid algebra K[M] is a domain if and only if K is a domain and M a regular torsion-free binoid.~~

~~• Proof. If K[M] is a domain, the binoid M has to be integral; that is, K[M] ∼= KM . The theorem follows now from Proposition 3.1.6.~~

~~Corollary 3.2.11. If M is a torsion-free binoid and K a domain, then K[M/M+] is a domain.~~

~~Proof. This is an immediate consequence of Theorem 3.2.10 since M/M = M × is regular and, + ∪{∞} by Lemma 2.3.8, torsion-free.~~

~~Here are two important examples of non-commutative algebras that can be realized as binoid algebras.~~

~~Example 3.2.12. (Matrix algebras) For V = 1,...,n , n 1, let M = e (i, j) V V { } ≥ { i,j | ∈ × }∪{∞} be the binoid with addition given by~~

ei,l , if j = k , ei,j + ek,l = ( , otherwise. ∞ If M (K) denotes the K -algebra of (n n)-matrices, then there is a K - algebra isomorphism n × ∼ K[M] M (K) ,T ei,j E −→ n 7−→ ij where Eij = (akl)1≤k,l≤n is the elementary matrix with akl = 1 if (k,l) = (i, j) and 0 otherwise. 3.2 Binoid algebras 101

Example 3.2.13. (Path algebras) A finite quiver Q is a finite directed graph possibly with multiple arrows and loops. In other words, Q = (V,A,s,t) is a quadruple consisting of the finite sets of vertices V and arrows A between them, and two maps s,t : A V , which associate to each arrow β A its → ∈ source s(β) V and its target t(β) V . Let a,b V be two vertices. A path from a to b in Q is a ∈ ∈ ∈ sequence (a β β b), where β A, i 1,...,ℓ , with s(β )= a and s(β )= t(β ) for 1 i<ℓ, | 1 ··· ℓ| i ∈ ∈{ } 1 i+1 i ≤ and t(β )= b, which may be briefly denoted by (β β ) or illustrated as follows ℓ 1 ··· ℓ β β β β a = a 1 a 2 a 3 ℓ a = b . 0 −→ 1 −→ 2 −→ · · · −→ ℓ The number ℓ is called the length of (β β ). Furthermore, we associate to each vertice a V a 1 ··· ℓ ∈ stationary path at a of lenght zero, denoted by ε = (a a). If Q denotes the set of all paths of lengh a || ℓ ℓ, then Q = V and Q = A. Let n = A . Consider the set 0 ∼ 1 ∼ | | M = (i ,...,i ) 1,...,n ℓ (β β ) Q ℓ { 1 ℓ ∈{ } | i1 ··· iℓ ∈ ℓ}∪{∞} of ℓ - tuples which belong to a path in Q with adjoint. Tuples with no corresponding path in Q will ℓ ∞ be identified with . Such tuples do not exist if and only if A = β and s(β) = t(β). Otherwise, ∞ { } the tuple has every length. It follows that ∞

MQ := Mℓ Mℓ≥0 is a binoid with respect to (i ,...,i ) (j ,...,j ) = (i ,...,i , j ,...,j ) if this tuple belongs to a 1 ℓ ◦ 1 k 1 ℓ 1 k path in Q and otherwise. The corresponding binoid algebra ℓ+k ∞

Q K[M ] = KXi1 Xiℓ • ··· (i1,...,iMℓ)∈MQ is given by the quotient K X ,...,X /KX∞ of the free associative K - algebra with non-commuting h 1 ni variables X ,...,X and KX∞ = ( X X (i ,...,i ) / M ,ℓ 0). If K[Q] denotes the path 1 n K i1 ··· iℓ | 1 ℓ ∈ ℓ ≥ algebra of Q with K - basis (β β ) Q ℓ> 0 , then the natural map given by (β β ) { i1 ··· iℓ ∈ ℓ | } i1 ··· ik 7→ X X is a K - algebra isomorphism i1 ··· ik ∼ K[Q] K[MQ ] . −→ In case K is an algebraically closed field, the preceding definition of the path algebra associated to a quiver agrees with the classical one from representation theory ([7, Chapter 4]). It is a well-known fact that K[Q] has a unit, namely 1 Q = ε , if and only if V is finite. In this case, ε a V is K[ ] a∈V a { a | ∈ } a complete set of primitive orthogonal idempotents for K[Q]. P It might be useful to have another, slightly different definition of a (finite) quiver changing this situation. For this, let (V,A,s,t) as before, but instead of associating to each vertice a stationary path, we consider only the empty (or totally stationary) path subject to the rule ∅ β = β = β ∅ ∅ for all β A (i.e. Q = ). Accordingly, we call a quiver defined in this way a q u i v e r w i t h 1 ∈ 0 {∅} (= ), and denote it by Q1. This definition makes even more sense when passing to the binoid algebra ∅ 1 of MQ1 because the unit element of the quiver coincides with the unit of the K - algebra K[Q ]. If V = a , then Q = Q1 with 1 = ε . { } a 102 3 Basic concepts of binoid algebras

~~3.3 Ideals in binoid algebras~~

~~Convention. In this section, arbitrary binoids are assumed to be commutative.~~

Recall that every ideal in a binoid M deﬁnes a monomial ideal in K[M], namely I K[ ] = KT a . I Ma∈I Conversely, to each ideal a K[M] there is an ideal of exponents in M, ⊆ (a) := a M T a a . I { ∈ | ∈ } Lemma 3.3.1. (1) Let and be two ideals in M. I J (a) K[ ] = K[ ]+ K[ ]. I ∪ J I J (b) K[ ] = K[ ] K[ ]. I ∩ J I ∩ J (c) K[ + ] = K[ ] K[ ]. I J I · J (2) K[ (a)] a for every ideal a in K[M]. Moreover, if a and b are monomial ideals in K[M], then I ⊆ (a) K[ (a)] = a. In particular, ( ) establishes a bijection between the set of ideals of M and I I − the set of monomial ideals of K[M] with inverse K[ ], namely K[ ] and (a) [ a. − I 7→ I I ← (b) b a if and only if (b) (a). ⊆ I ⊆ I (c) If a is a radical ideal then so is (a). I Proof. All assertions are easily veriﬁed.

~~Proposition 3.3.2. Let K be a ﬁeld. If M is positive, then K[M+] is a maximal ideal in K[M].~~

~~Proof. We have K = K[ 0, ] = K[M/M ] = K[M]/K[M ], where the latter identity is due to ∼ { ∞} + + Corollary 3.2.8(3).~~

This result fails for non-positive binoids. As an easy example consider the binoid group M = (Z/nZ)∞ with n 2. Then K[M]/K[M ] = K[M/M ] = K[M] = K[X]/(Xn 1) is not a field ≥ + + ∼ − since X 1 is a zero-divisor. In particular, K[M ]= K[ ] = 0 is not a maximal ideal in K[M]. − + {∞} Corollary 3.3.3. Let K be a field. If M is finitely generated and positive, then

~~n 1 dimK K[M]/(K[M+]) = H(n,M) .~~

~~Proof. We have~~

n a K[M]/(K[M+]) = K[M]/K[nM+] = K[M/nM+] = KT • a∈(M/nMM +) as K - vector spaces, hence dim K[M]/(K[M ])n = #(M/nM ) 1 = H(n,M). K + + − Proposition 3.3.4. Let M be a binoid. If p spec K[M], then (p) spec(M). In particular, every ∈ I ∈ monomial prime ideal in K[M] is of the form K[ ] for some prime ideal spec M. Conversely, if P P ∈ M is torsion-free and regular, spec M, and K a domain, then K[ ] is a (monomial) prime ideal P ∈ P in K[M].

~~1Here, we use the convention a0 := R for an ideal a in a ring R. 3.4 K[N] - modules 103~~

Proof. Let p spec K[M] and a,b M with a+b (p). Then T a+b = T a T b p, which implies that ∈ ∈ ∈ I · ∈ T a p or T b p by the prime property. Hence, a (p) or b (p). This shows that (p) is prime. ∈ ∈ ∈ I ∈ I I The additional observation follows now from Lemma 3.3.1(2a). Conversely, if is a prime ideal, then P M/ is integral and again torsion-free and cancellative by Lemma 2.1.19 and Lemma 2.1.20. Hence, P K[M/ ] = K[M]/K[ ] is a domain by Theorem 3.2.10, which implies that K[ ] spec K[M]. P ∼ P P ∈ Corollary 3.3.5. Let K be a domain and M a torsion-free cancellative binoid. If p is a minimal prime ideal of K[M], then (p) is a minimal prime in M. I Proof. By Proposition 3.3.4, only the minimality of (p) needs to be veriﬁed. So suppose that M I Q⊆ is a prime ideal with ( (p). Then K[ ] spec K[M] by Proposition 3.3.4 with K[ ] ( K[ (p)] = Q I Q ∈ Q I p, which contradicts the minimality of p. Hence, (p) is a minimal prime ideal. I

~~3.4 K[N] - modules~~

The concept of binoid algebras for binoids can be generalized to arbitrary N - sets (S,p); that is, to every N -set (S,p) one can associate a K[N] - module K[S]. The following deﬁnition of K[S] suggests what the subsequent two results show: many of the results on binoid algebras given in Section 3.2 can be generalized to N -sets and their associated K[N] - modules since every binoid is a 0, -set. In { ∞} this thesis, the results are not needed in their more general module theoretic form, but rather as they are stated in Section 3.2 for binoid algebras. However, there is a purely module theoretic result, cf. Proposition 3.4.7, even when the N - sets are given by binoids (i.e. N = 0, ). We close this section { ∞} by applying this result to an important example, the blowup binoid.

~~Definition 3.4.1. For an N -set (S,p) let K[S] denote the K[N] - module given by the set of all formal sums r Xs with T S• finite, r K, and scalar multiplication defined by s∈T s ⊆ s ∈~~

P a+s a s rarsX , if a + s = p , raX rsX := 6 · (0 , otherwise, where a N and s S. ∈ ∈ Example 3.4.2. (1) The ideal K[ ] is a K[M] - module that can be understood as coming from the M -set ( , ). I I ∞ (2) Let Q Zd be a submonoid and F a proper ﬁlter of Q∞. We consider the subset ⊆ F Q = f q f F, q Q − { − | ∈ ∈ } of Zd, and its translates a + F Q for a Zd. The pointed set M := (a + F Q)∞ is a Q∞ -set − ∈ a − with respect to the operation Q∞ M M , × a −→ a

q + u , if q + u Ma , (q,u) ∈ 7−→ ( , otherwise. ∞ The set F Q is called the injective hull of the ﬁlter F , and the K[Q∞] - module K[M ] an − a indecomposable injective of Q, cf. [41, Deﬁnition 11.7].

~~Lemma 3.4.3. S is a ﬁnitely generated N - set if and only if K[S] is a ﬁnitely generated K[N] - module.~~

~~Proof. This is clear. 104 3 Basic concepts of binoid algebras~~

~~Every K[N] - module V is also an N -set with respect to the (left) operation of N on (V, 0) given by~~

N V V, (a, v) a + v := Xa v . × −→ 7−→ · In particular, K[S] is again an N -set for every N -set (S,p), and unless K = 0 there is a canonical injective N - map ι : S K[S] , s Xs . S −→ 7−→ The associated K[N] - module of an N -set is a universal object and therefore unique up to isomorphism.

Proposition 3.4.4. Let (S,p) be an N - set and V a K[N] - module. Every N - map ϕ : (S,p) (V, 0) → gives rise to a unique K[N] - module homomorphism φ : K[S] V such that the diagram → ϕ S / V ③= ③③ ιS ③③ ③ φ ③③ K[S] commutes.

Proof. The map φ defined by φ( r Xs) := r ϕ(s) with r K, s T , and T S• finite, s∈T s s∈T s s ∈ ∈ ⊆ is a well-defined K - module homomorphism with φι = ϕ. To verify that φ is also a homomorphism P P S of K[N] - modules observe that ϕ(a + s)= a + ϕ(s)= Xa ϕ(s) for all a N and s S. Therefore, if · • ∈ ∈ F = r Xa K[N] and r Xs K[S] for finite sets A N and T S p , then a∈A a ∈ s∈T s ∈ ⊆ ⊆ \{ } P P φ F r Xs = φ r r Xa+s · s a s sX∈T a∈XA,s∈T = rarsϕ(a + s) a∈XA,s∈T a = rarsX ϕ(s) a∈XA,s∈T = F r ϕ(s) · s sX∈T = F φ r Xs . · s sX∈T Corollary 3.4.5. Given an N - map ϕ : S T of N -sets (S,p) and (T, q), there exists a unique → K[N] - module homomorphism φ : K[S] K[T ] such that the diagram → ϕ S / T

~~ιS ιT K[ϕ] K[S] / K[T ] commutes. In particular, there is a binoid homomorphism~~

(map S, , id , ϕ ) (End K[S], , id , 0 ) p ◦ S ∞ −→ K ◦ K[S] K[S] with ϕ K[ϕ], where End K[S] denotes the ring of all K - module homomorphisms K[S] K[S] 7→ K → and 0K[S] the zero map. 3.4 K[N] - modules 105

Proof. This is an immediate consequence of Proposition 3.4.4 applied to the N - map given by the composition ιT ϕ : S T K[T ]. In particular, K[ϕ]: K[S] K[T ] is given by K[ϕ](F ) = ϕ(s) → → s ′ → ′ ′ r X for F = ′ r X K[S] with r K, s S , and S S ﬁnite. This shows that s∈S s s∈S s ∈ s ∈ ∈ ⊆ K[ϕ ψ]= K[ϕ] K[ψ], K[id ] = id , and K[ϕ ]=0 , which implies the supplement. P ◦ ◦ P S K[S] ∞ K[S] Proposition 3.4.6. Let (S,p) be an N - set. K[S] is a K[N] - module such that there is a commutative diagram ϕ N / mapp S

~~ιN K[-]~~

φ K[N] / EndK K[S] , where ϕ is a binoid homomorphism and φ a ring homomorphism. Proof. The operation N S S, (a,s) a + s, induces a canonical operation K[N] K[S] K[S] × → 7→ × → generated by (r T a, r T s) r r T a+s. Thus, there is a ring homomorphism φ : K[N] End K[S] a s 7→ a s → K with F (φ(F ): G G F ), which makes the diagram commutative. 7→ 7−→ ·

~~Proposition 3.4.7. Given a family (Si,pi)i∈I of N - sets, there is a K[N] - module isomorphism:~~

K S = K[S ] . · i ∼ i h i[∈I i Mi∈I ,(Proof. By Corollary 3.4.5, the family (ϕ ) of injective N - maps S ֒ S =: S with s (s; i i i∈I i → · i∈I i 7→ i I, induces a family ι : K[S ] K[S], i I, of K[N] - module homomorphisms, which gives rise ∈ i i → ∈ S to a K[N] - module homomorphism

φ : K[S ] K[S] i −→ Mi∈I with φ((F ) ) = ι (F ), cf. [51, Satz 39.1]. On the other hand, the surjections S S , i I, i i∈I i∈I i i → i ∈ with (s; k) (s; i) if k = i and p otherwise, yield by Corollary 3.4.5 K[N] - module homomorphisms 7→ P i K[S] K[S ], i I, which induce a K[N] - module homomorphism → i ∈ ψ : K[S] K[S ] −→ i Mi∈I a a with raX raX , where A Si is a finite set. It is easily checked that a∈A 7→ i∈I a∈Si∩A ⊆ · i∈I φ and ψ are inverse to each other. P P P S Remark 3.4.8. For a binoid M and an ideal M define the blowup binoid R of to be the I ⊆ I I pointed union of the family (n , ) of pointed sets, i.e. I ∞ n≥0 R := n , I · I n≥0 with addition defined by [ (f; n) + (g; m) := (f + g; n + m) .

The identity element is then given by (0; 0) and the absorbing element by the element that arises by glueing the elements ( ; n), n 0, together. Thus, R is generated by M =0 and =1 , and ∞ ≥ I · I I · I can be considered as a subbinoid of M N∞ with respect to (a; n) a n. By Proposition 3.4.7, it ∧ 7→ ∧ follows that the binoid algebra of the blowup binoid

K[R ] = K[n ] = K[ ]n I I I Mn≥0 Mn≥0 106 3 Basic concepts of binoid algebras is the blowup algebra of K[ ] K[M], cf. [23, Chapter 5.2]. Moreover, (n , ) is an M -set for I ⊆ I ∞ every n 0 and R is an M - binoid via the embedding M ֒ R , a (a; 0). Hence, we have a ≥ I → I 7→ commutative diagram

RI > ❇ ⑦⑦> ❇❇ ι ⑦⑦ ❇❇ ⑦⑦ ❇❇ ⑦ ❇❇ ⑦⑦ M R (M/ ) = n (n + 1) , ❅ I M ∼ · ❅❅ ∧⑥> I n≥0 I I ❅ ⑥⑥ ❅❅ ⑥ S π ❅ ⑥⑥ ❅❅ ⑥ ⑥⑥ M/ I where the isomorphism is given as M - binoids as one easily checks (where the addition on the pointed union is deﬁned as for R ), cf. Proposition 1.10.7. The M - binoid n (n + 1) =: gr ( ) is I · n≥0 I I M I called the associated graded binoid of the binoid M with respect to the ideal M. The S I ⊆ terminology is justiﬁed because for the binoid algebra we obtain

K[R (M/ )] = K[R ] K[M/ ] I ∧M I ∼ I ⊗K[M] I = K[ ]n K[M] K[ ] ∼ I ⊗K[M] I n≥0 M = K[ ]n K[M] K[ ] ∼ I ⊗K[M] I n≥0 M = K[ ]n K[ ]n+1 ∼ I I n≥0 M =: grK[M](K[I]) , where grA(a) denotes the associated graded ring of the ring A with respect to the (ring) ideal a ( A, cf. [23, Chapter 5.1]. An important case is when = M . If M is positive, then I + gr (K[M ]) = K[R ] K, K[M] + M+ ⊗K[M] where K is a K[M]-algebra with respect to K[M] K, T a 1 if a = 0 and 0 otherwise. → 7→

~~∞ ∞ Example 3.4.9. Consider the integral binoid M := N≥2 0 N . The blowup binoid of M with ∞ ∪{ }⊆ respect to M+ = N≥2 is given by~~

R = nM = N∞ . M+ · + · ≥2n n[≥0 n[≥0 Since nM/(n+1)M = N∞ /N∞ = 2n, 2n+1 ∞, we have (2n+1; n)+(2n+1; n)=(4n+2;2n)= ≥2n ≥2n+2 { } though (2n + 1; n) = in ∞ 6 ∞ gr (M ) = (2n; n), (2n + 1; n) ∞ . M + · { } n[≥0

~~Thus, the associated graded binoid grM M+ is not reduced, hence not integral. In particular,~~

~~• K[grM (M+)] = grKM • (M+) is a binoid algebra that is not an integral domain. 3.5 Binoid algebras of N - binoids 107~~

~~3.5 Binoid algebras of N - binoids~~

~~Convention. In this section, N always denotes a commutative binoid.~~

By Remark 3.2.5(2), the structure homomorphism ϕ : N M of an N - binoid M induces a ring → homomorphism K[ϕ]: K[N] K[M]. This deﬁnes a K[N]-algebra structure on K[M]. → Corollary 3.5.1. Every homomorphism ϕ : M M ′ of N - binoids induces a unique K[N] - algebra → • homomorphism φ : K[M] K[M ′] with φ(rT a)= rT ϕ(a), r K, a M . → ∈ ∈

~~Proof. This is clear by Corollary 3.2.4 (with K = L and α = idK ) and the commutative diagram~~

~~N ⑦ ❈❈ ψ ⑦⑦ ❈❈ ψ′ ⑦⑦ ❈❈ ⑦⑦ ❈❈ ~⑦ ϕ ! M / M ′ , where ψ and ψ′ are the structure homomorphisms of M and M ′, respectively.~~

~~The following corollary can be generalized to N - sets, which we will not need here.~~

~~Corollary 3.5.2. Let (Mi)i∈I be a ﬁnite family of commutative N - binoids. Then~~

K M = K[M ] N i ∼ K[N] i h i^∈I i Oi∈I as K - algebras. In particular, K M K[M ] . i ∼= K i h i^∈I i Oi∈I Proof. Using an inductive argument, it suﬃces to prove the statement for I = 1, 2 . So let M1 =: M ′ { } and M2 =: M be commutative binoids. There is a commutative diagram of K - algebras,

K[M] ❘❘ ♥♥7 ❘❘❘ ♥♥♥ ❘❘❘ϕ ♥♥ ι ❘❘❘ ♥♥♥ ❘❘❘ ♥♥ ❘) K[N] K[M] K[M ′] K[M M ′] P ⊗K[N] ∧N PPP O ❧❧❧5 PP ′ ❧❧ P ι ❧❧ PP ❧ ′ PP ❧❧❧ ϕ P' ❧❧❧ K[M ′] where ι and ι′ are the canonical inclusions, and ϕ and ϕ′ are the K[N] - algebra homomorphisms induced by the inclusions M ֒ M M ′ and M ′ ֒ M M ′, cf. Corollary 3.5.1. By the universal → ∧N → ∧N property of the tensor product, cf. [51, Satz 80.9], we have a K - algebra homomorphism

ψ : K[M] K[M ′] K[M M ′] ⊗K[N] −→ ∧N with ψ(F G) = ϕ(F )ϕ′(G), which is even a K[N] - algebra homomorphism since ϕ and ϕ′ are ⊗K[N] so. On the other hand, the well-deﬁned binoid homomorphism

′ M M ′ K[M] K[M ′] with a a′ T a T a ∧N −→ ⊗K[N] ∧N 7−→ ⊗K[N] ′ ′ induces by Proposition 3.2.3 a K - algebra homomorphism K[M N M ] K[M] K[N] K[M ] with ′ ∧ → ⊗ r(a a′) r(T a T a ), which is also a K[N] - algebra homomorphism and inverse to ψ. ∧N 7→ ⊗K[N] 108 3 Basic concepts of binoid algebras

~~Corollary 3.5.3. If M is a ﬁnitely generated commutative N - binoid, then K[M] is a ﬁnitely generated K[N] - algebra.~~

Proof. Let x ,...,x be a generating set of M. By Remark 3.2.5(2) and Corollary 3.5.2, the binoid { 1 r} epimorphism ϕ : N (Nr)∞ M, ∧ −→ a (n ,...,n ) ϕ(a)+ n x + + n x , induces a K - algebra epimorphism ∧ 1 r 7→ 1 1 ··· r r K[ϕ]: K[N][X ,...,X ] K[M] , 1 r −→ with rT aXν rT ϕ(a)Xν, ν = (n ,...,n ). Obviously, K[ϕ] is a K[N] - algebra homomorphism. This 7→ 1 r proves the statement.

Remark 3.5.4. (1) The supplement of Corollary 3.5.2 can be obtained from earlier results when taking extra assumptions on the binoids into account. If, for instance, M and M ′ are integral, then M M ′ = • • ∧ ∼ (M M ′ )∞ by Lemma 1.8.8(3). Therefore, × ′ • ′• K[M M ] ∼= K(M M ) ∧ • × • = KM KM ′ ∼ ⊗K = K[M] K[M ′] , ∼ ⊗K where the isomorphism in the middle is due to Corollary 3.1.5(4). However, for arbitrary (not necessarily integral) commutative binoids this also follows from Corollary 3.2.8(3) since M M ′ = ∧ ∼ (M M ′)/ with = (a,b) a = or b = , which yields × I I { | ∞ ∞} K[M M ′] = K(M M ′)/(T c c ) ∧ ∼ × | ∈ I = (KM KM ′)/(KT ∞ KM ′ + KM KT ∞) ∼ ⊗K ⊗K ⊗K = (KM/(T ∞)) (KM ′/(T ∞)) ∼ ⊗K = K[M] K[M ′] . ∼ ⊗K The isomorphism K[(M M ′)/ ] = K[M] K[M ′] can already be found in [46, Chapter 4, × I ∼ ⊗K Lemma 10]. (2) Corollary 3.5.2 has shown that the tensor product of binoid algebras comes from the smash product of the involved binoids, while for the product we get

~~K[M M ′] = K(M M ′)/(T (∞,∞)) × × = KM KM ′/(U ∞ V ∞) , ∼ ⊗K ⊗K cf. Corollary 3.1.5(4). By Proposition 3.2.3, there is a commutative diagram~~

~~ιM×M′ M M ′ / K[M M ′] × ♠♠ × ♠♠♠ ι:=(ιM ,ιM′ ) ♠♠ ♠♠♠ ˜ι v♠♠ K[M] K[M ′] ×~~

a a′ ′ ′ (a,a′) with ˜ιιM×M ′ = ι. Thus, (U , V )= ι(a,a )=˜ιιM×M ′ (a,a )=˜ι(T ). Clearly, ˜ι is surjective, but it is not injective if M ′ and M are nonzero, since then ˜ι(T (0,0))=(1, 1)=˜ι(T (0,∞) + T (∞,0)) while T (0,0) = T (0,∞) + T (∞,0). 6 3.5 Binoid algebras of N - binoids 109

~~′ ′ ′ ′ (3) The natural binoid epimorphisms π∧ : M M M M and π : : M M M : M , cf. × → ∧ ∧ → ∪ Remark 1.9.6, induce by Corollary 3.2.4 the K - algebra epimorphisms∪~~

K[π∧] K[π : ] K[M M ′] K[M M ′] K[M : M ′] . × −→ ∧ −→∪ ∪ If M and M ′ are commutative, then M : M ′ = (M M ′)/J with J = a a′ a = 0 and ∪ ∧ { ∧ | 6 a′ =0 , cf. Example 2.1.18. By Corollary 3.2.8(3), this gives 6 } ′ K[M : M ′] = K[M] K[M ′]/(U a V a a,a′ = 0) . ∪ ⊗K ⊗ | 6 Example 3.5.5. We apply the results of the preceding remark to M = M ′ = N∞. This yields the following K - algebra isomorphisms:

K[N∞ N∞] = KN∞ KN∞/(U ∞ V ∞) × ⊗K ⊗K = (K K[X]) (K K[Y ])/((1, 0) (1, 0)) ∼ × ⊗K × ⊗ = K[X] K[Y ] K[X, Y ] ∼ × × K[N∞ N∞] = K[X] K[Y ] = K[X, Y ] ∧ ∼ ⊗K ∼ K[N∞ : N∞] = K[X] K[Y ]/(X Y ) = K[X, Y ]/(XY ) , ∪ ∼ ⊗K ⊗ ∼ where K[X], K[Y ], and K[X, Y ] denote the polynomial rings with indeterminates X and/or Y .

~~4 Topology of commutative binoids~~

In this chapter, we investigate the spectrum and the K - spectrum, K a ﬁeld, of a commutative binoid as topological spaces. The topology on the spectrum is deﬁned analogous to the Zariski topology on the spectrum of a ring. All results resemble those from ring theory, and the same applies to the dimension theory of binoids, with which we deal in the third section. We introduce the booleanization of a binoid M, which is a fairly easier binoid to study, and show that its spectrum is homeomorphic to that of M. Similar to a result on characters of semigroups, we give a criterion when the K - points of a binoid M separate the elements of M, cf. Proposition 4.4.11. The topology on K-spec M comes from that on K-Spec K[M], and the description of its properties will be continued in the next chapter. Here we mainly focus on connectedness properties, in particular of hypersurfaces, and show that K-spec M is the union of its cancellative components, which serve as the key tool to prove our main criterion for connectedness, cf. Theorem 4.5.2. Convention. In this chapter, arbitrary binoids are assumed to be commutative unless otherwise stated.

~~4.1 Spectrum~~

In this section, we define a topology on the spectrum of a binoid M similar to the Zariski topology on the spectrum of a ring R. An excellent description of the latter can be found in [47, Chapter 3.A], which we will partly follow since many results translate directly to binoids and their spectra. The reader may also consult [31, Chapter I 1], [42, Chapter II 1], or [9, Chapter II 4.1] for spectra of § § § rings. The spectrum of a binoid equipped with the Zariski topology and its subspace consisting of all minimal prime ideals have been studied in detail in [37]. See also [19]. Recall, cf. Section 2.2, that the semibinoid (spec M, ,M ) is a binoid with identity element ∪ + {∞} if M is integral. However, one can always turn (spec M, ,M ) into a binoid by adjoining as an ∪ + {∅} identity element irrespectively of the integrality of M. Definition 4.1.1. Let M be binoid. The commutative binoid (spec M , , ,M ) is called the ∪ {∅} ∪ ∅ + extended spectrum of M, which will be denoted by spec∅ M. Corollary 4.1.2. For M = , the boolean binoids spec∅ M, M v , and (M) are isomorphic. 6 {∞} F ∩ Proof. Extending the maps from Corollary 2.2.6 to these binoids by (χ =) α M gives the M ∅ ↔ ∅ ↔ well-defined isomorphisms.

~~Deﬁnition 4.1.3. For a subset A M let ⊆ V(A) := spec M A . {P ∈ | ⊆ P} The complements in spec M and spec∅ M are denoted by~~

D(A) := spec M V(A)= spec M A and D∅(A) := spec∅ M V(A)=D(A) . \ {P ∈ | 6⊆ P} \ ∪ {∅} If A = f , we shall write V(f) instead of V( f ) and the same for the complements D(f) and D∅(f). { } { } 112 4 Topology of commutative binoids

Lemma 4.1.4. Let M be a binoid, f,g M, and A, A ,B M, i I, subsets. ∈ i ⊆ ∈ (1) i∈I V(Ai)=V( i∈I Ai). (2) V(f) V(g)=V(f + g). T ∪ S (3) If A B, then V(B) V(A). ⊆ ⊆ (4) V(f)= if and only if f M ×. ∅ ∈ (5) V(f) = spec M if and only if f is nilpotent. (6) V(A)=V( A )=V( A ). Mh i Mh i Proof. (1)-(4) are straightforwardp and (5) follows immediatley from Corollary 2.3.10. (6) Set A =: Mh i . We always have A √ , and hence V(√ ) V( ) V(A) by (3). For the other inclusions, A ⊆A⊆ A A ⊆ A ⊆ it suﬃces to show that V(A) V(√ ). Given V(A), we have to verify that √ . So let ⊆ A P ∈ A⊆P f √ . This means nf for some n 1, which implies that nf = a + m with a A and ∈ A ∈ A ≥ ∈ ⊆ P m M. In particular, nf , and therefore f by the prime property. ∈ ∈P ∈P By taking complements, Lemma 4.1.4 translates to:

Lemma 4.1.5. Let M be a binoid, f,g M, and A, A ,B M, i I, subsets. ∈ i ⊆ ∈ (1) i∈I D(Ai)=D( i∈I Ai). (2) D(f) D(g)=D(f + g). S ∩ S (3) If A B, then D(A) D(B). ⊆ ⊆ (4) D(f) = spec M if and only if f M ×. ∈ (5) D(f)= if and only if f is nilpotent. ∅ (6) D(A)=D( A ) =D( A ). Mh i Mh i Proof. All statements are immediatep consequences of Lemma 4.1.4.

The preceding lemmata show that spec M and spec∅ M are topological spaces for every binoid M, where the closed sets are given by V(A), A M. In either case this topology will be called the ⊆ ∅ Zariski topology. The complements D(A) and D (A) are the open sets, where (D(f))f∈M and ∅ ∅ (D (f))f∈M deﬁne a basis of open sets of the Zariski topology on spec M and spec M, respectively. From now on, when we consider the (extended) spectrum of a binoid as a topological space, we mean the (extended) spectrum together with the Zariski topology. Remark 4.1.6. (1) There are two natural topologies on a ﬁnite poset (X, ): the lower topology, where the open ⊆ sets are given by the subset-closed sets of X. In other words,

D X open : (A D and B A B D) ⊆ ⇔ ∈ ⊆ ⇒ ∈ (which means that the closed sets are given by the superset-closed sets), and the upper topology, where the open sets are given by the superset-closed sets of X. In other words,

D X open : (A D and A B B D) . ⊆ ⇔ ∈ ⊆ ⇒ ∈ Thus, if spec M is a finite set, for instance when M is finitely generated, then the Zariski topology on spec M coincides with the lower topology on (spec M, ). ⊆ (2) By Corollary 2.2.6, spec M and (M) M are isomorphic as semigroups by taking the com- F \{ } plements. Hence, considering a topology on spec M or on (M) is essentially the same. Those F topological spaces that arise as the filtrum of a commutative monoid have been characterized in [10, Satz 2.3.2]. 4.1 Spectrum 113

(3) When dealing with monoids the empty set is usually considered as a prime ideal to ensure that the spectrum is never empty, which were the case if M is a group. By this convention, the set of all minimal prime ideals of M is always a singleton. In particular, the spectrum admits always a unique generic point and is irreducible, cf. [45, Chapter 1.4], which still holds true if one adjoins an absorbing element to the monoid, cf. Corollary 4.1.17.

Corollary 4.1.7. Let M be a binoid. (1) If is an ideal in M, then I spec(M/ ) = V( ) spec M. I ∼ I ⊆ (2) If S M is a submonoid of M, then ⊆ spec M = spec M S = . S ∼ {P ∈ | P ∩ ∅} If, in addition, S can be generated by a ﬁnite set A S, then ⊆ spec M = D(f) spec M, S ∼ ⊆ where f = g. In particular, spec M = D(f) for every f M. g∈A f ∈ (3) The following statements are equivalent for spec M: P P ∈ (a) M = M for some f M. P f ∈ (b) spec M = D(f) for some f M. P ∈ (c) spec MP is open in spec M.

Proof. (1) and the first part of (2) are just restatements of Corollary 2.2.10 and Corollary 2.2.17. If S is generated by the finite set A S, we obtain from the first part and Lemma 4.1.5(2) that ⊆ spec M = spec M g for all g A = D(g) = D(f) S ∼ {P ∈ | 6∈ P ∈ } g\∈A with f = g. (3) The implication (a) (b) is clear by (2), and (b) (c) is trivial. So let g∈A ⇒ ⇒ spec M be open in spec M. Then there is a subset B M such that spec M = D(B)= D(f). P P ⊆ P f∈B On the other hand, spec M = spec M by (2). Hence, D(f) for some f B, but P {Q ∈ |Q⊆P} P ∈ S ∈ then spec M D(f), which implies that spec M = D(f). P ⊆ P The equivalences of the following proposition will become very useful in Section 4.2.

Proposition 4.1.8. Let M be a binoid and f,g M. The following statements are equivalent: ∈ (1) V(g) V(f). ⊆ (2) D(f) D(g). ⊆ (3) Filt(g) Filt(f). ⊆ (4) nf = g + x for some n N and x M; that is, f g . ∈ ∈ M h i⊆ M h i In particular, for a boolean binoid M, one has D(f)=D(g) ifp and only if f = g.

~~Proof. Clearly, (1) and (2) are equivalent. So assume that D(f) D(g) for f,g M. By taking ⊆ ∈ complements of the prime ideals in these basic open sets, we get~~

F := H (M) f H H (M) g H =: G, { ∈F | ∈ }⊆{ ∈F | ∈ } which implies that Filt(g)= H H = Filt(f). Thus, (3) follows from (2). The implica- H∈G ⊆ H∈F tion (3) (4) was observed in Remark 1.4.37. (4) (1) is obvious. The if part of the supplement is ⇒ T T ⇒ 114 4 Topology of commutative binoids trivial (and holds for arbitrary binoids). So let M be boolean and D(f) = D(g) for some f,g M. By ∈ the equivalence of (2) and (4), there are elements x, y M with f = g + x and g = f + y. It follows ∈ f = f + f = f + g + x = f + g + g + x = f + g + f = f + g and by symmetry g = f + g. Thus, f = g.

Remark 4.1.9. If A is a subset such that Filt(A), which is the smallest filter containing A, is a proper filter, then M Filt(A) =: is a prime ideal in M with A . Since every prime ideal \ PA 6⊆ PA is the complement of a (proper) filter by Corollary 2.2.6, is the largest prime ideal with A . PA 6⊆ PA Therefore, D(A) and for every D(A), or in other words = . PA ∈ P⊆PA P ∈ PA P∈D(A) P Lemma 4.1.10. Let U spec M be an open set, , ′ spec M, and f M. S ⊆ P P ∈ ∈ (1) If U, then ′ implies ′ U. P ∈ P ⊆P P ∈ (2) M Filt(f) U if and only if D(f) U. \ ∈ ⊆ (3) If D(f) U is an open cover, then D(f) U for one i I. ⊆ i∈I i ⊆ i ∈ Proof. (1) SinceSU is open, there is an A M such that U = D(A)= spec M A . Then ⊆ {P ∈ | 6⊆ P} A ′ if ′ U. The assertions (2) and (3) are clear if f is nilpotent, since then D(f) = 6⊆ P P ⊆ P ∈ ∅ and V(f) = M by Lemma 4.1.5(5) and Lemma 4.1.4(5). On the other hand, if f nil(M), then 6∈ Filt(f) = M. By Remark 4.1.9, D(f)= spec M , where := M Filt(f). This proves 6 {P ∈ |P ⊆Q} Q \ (2) and (3).

Corollary 4.1.11. Let M be a binoid. The spaces spec M and spec∅ M are quasi-compact and, if M = , connected. Moreover, the Zariski topology satisfies the separation axiom T . 6 {∞} 0 Proof. The first two properties are immediate consequences of Lemma 4.1.10(3) using the identifications spec M = D(0) and spec∅ M = D∅(0). To show the separation axiom let , spec M with P Q ∈ = . Choose f with f . Then D(f) is an open neighborhood of with D(f). This P 6 Q ∈ P 6∈ Q Q P 6∈ also proves the statement for spec∅ M.

For a subset E in a topological space X, the closure E of E with respect to the topology on X is the smallest closed subset of X containing E. In spec M, there is an easy description of these closures.

Deﬁnition 4.1.12. Given a subset E spec M, we denote the ideal by J(E). ⊆ P∈E P Proposition 4.1.13. E = V(J(E)) for every subset E spec M. T ⊆ Proof. For the inclusion , take an arbitrary E. By deﬁnition, we have J(E) , which implies ⊆ P ∈ ⊆P that V( ) V(J(E)) by Lemma 4.1.4(3). Now the inclusion follows since V( ). For the other P ⊆ P ∈ P inclusion, we need to show that every closed subset V spec M with E V contains V(J(E)). If ⊆ ⊆ V spec M is a closed subset, then V = V( ) for some ideal M. If, in addition, E V , then ⊆ I I ⊆ ⊆ for all E. Hence, J(E). Now V(J(E)) V follows from Lemma 4.1.4(3). I⊆P P ∈ I ⊆ ⊆ Since the intersection of radical ideals is again a radical ideal, J(E) is a radical ideal for every E spec M. Proposition 2.3.9 now translates to: ⊆ Corollary 4.1.14. J(V( )) = √ for every ideal M. In particular, the inclusion reversing I I I ⊆ assignments A J(A) and V( ) [ 7−→ I ←− I are inverse bijections between the closed subsets of spec M and the radical ideals of M.

~~Proof. The equality is just a restatement of Proposition 2.3.9 and implies V( ) J(V( )) = . I 7→ I 7→ I I Conversely we have A J(A) V(J(A)) = A by Proposition 4.1.13. 7→ 7→ 4.1 Spectrum 115~~

Recall that a topological space X is called i r r e d u c i b l e if X = and any two nonempty open 6 ∅ subsets of X intersect. Equivalently, X = and X is not the union of two proper closed subsets 6 ∅ of X. If X is an irreducible topological space with X = x , x X, then x is called a g e n e r i c { } ∈ point of X. A maximal irreducible subset of a topological space X with respect to is called an ⊆ irreducible component of X. To study the irreducible subsets and components of spec M, we need the following results.

Lemma 4.1.15. Let X be a topological space. (1) A subset Y X is irreducible if and only if its closure Y is irreducible. ⊆ (2) Every irreducible subset of X is contained in an irreducible component of X.

~~Proof. See [47, Proposition 3.A.10 and Proposition 3.A.13].~~

The ﬁrst statement of Lemma 4.1.15 shows that every irreducible component of X is closed, and by the second, we have that X is the union of its components since every point x , x X, of a { } ∈ topological space X is irreducible. The description of the irreducible components of spec M follows from the next result.

~~Proposition 4.1.16. The closed irreducible subsets of spec M are the sets V( ), spec M. In P P ∈ particular, every closed irreducible subset of spec M has a unique generic point.~~

Proof. The irreducibility of V( ) follows from Lemma 4.1.15 since every point in a topological space P is irreducible. Conversely, we have to show that every closed irreducible subset is of this form. If E is such a subset of spec M, then E = V( ) = for a radical ideal = M by Corollary 4.1.14. To verify I 6 ∅ I 6 the prime property of assume that f + g for some f,g M. If is a prime ideal containing I ∈ I ∈ P , then f or g . Thus, V( ) = V( f ) V( g ), which implies that I h i∪I ⊆P h i∪I ⊆P I h i ∪ I ∪ h i ∪ I V( ) = V( f ) or V( ) = V( g ) by the irreducibility of V( ). Hence, √ = f or I h i ∪ I I h i ∪ I I I h i ∪ I √ = g by Corollary 4.1.14. Thus, f or g . This proves that is a prime ideal. By I h i ∪ I ∈ I ∈ I I p Proposition 4.1.13, V( ) = V(J( )) = , which shows that the closed irreducible subset V( ) p P {P} {P} P has a generic point.

~~Corollary 4.1.17. The irreducible components of spec M are given by the sets V( ) with min M. P P ∈ Proof. This is clear by Proposition 4.1.16 and Lemma 4.1.4(3).~~

By Corollary 2.2.8, every binoid homomorphism ϕ : M N induces a map ϕ∗ : spec N spec M → → with ϕ−1( ). This signiﬁcant, albeit elementary result translates to the category Top of topo- P 7→ P logical spaces.

Proposition 4.1.18. Given a binoid homomorphism ϕ : M N, the induced semigroup homomor- → phism ϕ∗ : spec N spec M, ϕ−1( ), is continuous, namely → P 7→ P (ϕ∗)−1(D(A)) = D(ϕ(A)) and (ϕ∗)−1(V(A)) = V(ϕ(A)) for all A M. In particular, spec : comB Top is a contravariant functor from the category of ⊆ → commutative binoids into the category of topological spaces.

Proof. To prove that ϕ∗ is continuous, it suffices to consider the basic open sets D(f), f M, of the ∈ topology. For such a set, we obtain (ϕ∗)−1(D(f)) = spec N ϕ−1( ) D(f) . This proves the {P ∈ | P ∈ } first equality because ϕ−1( ) D(f) is equivalent to ϕ(f) by the definition of D(f). The second P ∈ 6∈ P identity for the closed subsets follows by taking complements. 116 4 Topology of commutative binoids

Corollary 4.1.19. If ϕ : M N is binoid epimorphism, the semigroup homomorphism ϕ∗ : → spec N spec M is a continuous embedding on im ϕ∗ V(ker ϕ). If N = M/ for an ideal M, → ⊆ I I ⊆ then spec M/ = V( ) a semigroups and as topological spaces, where V( ) carries the induced subspace I ∼ I I topology.

Proof. The continuous embedding ϕ∗ is given by Proposition 4.1.18. Since ker ϕ ϕ−1( )= ϕ∗( ) ⊆ P P for all spec N, we have im ϕ∗ V(ker ϕ). The statement for the special case N = M/ for an P ∈ ⊆ I ideal M, and ϕ = π : M M/ follows from Corollary 2.2.10. I ⊆ → I Example 4.1.20. Let M be a binoid.

(1) The semigroup isomorphism spec M ∼= spec Mred given in Corollary 2.2.11 can also be deduced from Corollary 4.1.19 since Mred = M/ nil(M), cf. Example 2.1.18, and V(nil(M)) = spec M by Lemma 4.1.4. Furthermore, this is an isomorphism of topological spaces by Corollary 4.1.19. c Similarly, we obtain spec Mint ∼= V(int (M)) as semigroups and as topological spaces from the c identiﬁcation Mint = M/ int (M). (2) Let N be an integral binoid. Every binoid homomorphism ϕ : M N factors through M/ , → Q where is the prime ideal ker ϕ = ϕ−1( ). In particular, ϕ factors through every minimal Q {∞N } prime ideal , which implies that ϕ∗ : spec N spec M factors through the irreducible P ⊆Q → component π∗ (spec M/ )=V( ) of spec M since P P P

ϕ ϕ∗ M / N induces spec M o spec N. = ③③ O qq ③ ∗ qq π ③③ π qq ③ ϕ¯ qq ϕ¯∗ ③③ xqq M/ spec M/ P P (3) Let M = . If M is reduced, there is a binoid embedding M M/ by Corollary 6 {∞} → P∈min M P 2.3.13. In other words, two elements of a reduced binoid M coincide if and only if they coincide Q on every irreducible component of M.

~~4.2 Booleanization~~

In this section, we introduce the booleanization of a binoid M which can be deﬁned for all, not necessarily commutative, binoids and is in either case a universal object. However, for a commutative binoid M, the booleanization has an explicit realization in terms of the basic open sets D(f), f M, ∈ of the Zariski topology on spec M, which we described at the end of the last section. For the moment we consider not necessarily commutative binoids but return very soon to the commutative situation.

Deﬁnition 4.2.1. Let M be an arbitrary (not necessarily commutative) binoid and the con- ∼bool gruence generated by f f + f ∼bool for f M. The binoid M/ =: M together with the canonical projection π : M M ∈ ∼bool bool bool → bool is called the b o o l e a n i z a t i o n of M.

If X is a generating set of the (not necessarily commutative) binoid M, then every element f M ∈ can be written as f = n x + + n x with (x , x ,...,x ) Xr, n 1, i 1,...,r , r N, such 1 1 ··· r r 1 2 r ∈ i ≥ ∈{ } ∈ that x = x for i 1,...,r 1 . Then π (f)= x + + x . i 6 i+1 ∈{ − } bool 1 ··· r The booleanization is a universal object as the following proposition shows. 4.2 Booleanization 117

Proposition 4.2.2. Let M be a (not necessarily commutative) binoid. Mbool is a boolean binoid such that nil(M) ker π , and whenever ϕ : M B is a binoid homomorphism with B boolean, there ⊆ bool → exists a unique binoid homomorphism ϕ¯ : M B such that the diagram bool → ϕ M / B ②< ②② πbool ② ②②ϕ¯ ②② Mbool commutes; that is, ϕπ¯ bool = ϕ. Proof. By deﬁnition, M is a boolean binoid such that f nf for all n 1. In particular, bool ∼bool ≥ nil(M) ker π . The existence of the induced binoid homomorphism follows from Lemma 1.6.3 ⊆ bool since . Indeed, this need only be checked for the generating relations f f + f, for ∼bool ≤∼ϕ ∼bool which ϕ(f)=2ϕ(f)= ϕ(f + f) obviously holds.

The congruence can be characterized more explicitly in the commutative situation. ∼bool Lemma 4.2.3. For f,g M, the following statements are equivalent: ∈ (1) f g. ∼bool (2) There are n,m N and x, y M such that nf = g + x and mg = f + y. ∈ ∈ (3) D(f) = D(g). (4) V(f)=V(g). (5) f = g . Mh i Mh i (6) Filt(p f) = Filt(p g). Proof. By Proposition 4.1.8, only the equivalence of (1) and (2) need to be shown. To prove (1) (2) ⇒ one easily checks that the relation on M deﬁned by f g, f,g M, if (2) is satisﬁed is a congruence ∼ ∼ ∈ with . Conversely, assume that nf = g + x and mg = f + y for some n,m N and x, y M. ∼bool ≤∼ ∈ ∈ Similar to the proof (of the supplement) of Proposition 4.1.8, we get f f + g and g f + g, ∼bool ∼bool hence f g. ∼bool From now on we focus again on commutative binoids. Note that in this case the we the equality nil(M) = ker π by the characterization of in (2) above. bool ∼bool Corollary 4.2.4. Given a boolean binoid B, there is a canonical binoid embedding

~~B P(spec B) with f D(f) . −→ ∩ 7−→ Proof. This is an immediate consequence of Lemma 4.2.3.~~

By applying the preceding result to the binoids P(V )∩ and P(V )∪ for a ﬁnite set V , we obtain the embeddings P(V ) P(spec P(V ) ) , J D(J)= I J ∩ −→ ∩ ∩ 7−→ {PI,∩ | ⊆ } and P(V ) P(spec P(V ) ) , J D(J)= J I , ∪ −→ ∪ ∩ 7−→ {PI,∪ | ⊆ } cf. Example 2.2.7. Now we give an explicit topological description of the booleanization. Corollary 4.2.5. Let M be a binoid. The set (M)= D(f) f M deﬁnes a commutative boolean B { | ∈ } binoid, namely ( (M), , spec M, ) =: (M) , B ∩ ∅ B ∩ which is isomorphic to Mbool. 118 4 Topology of commutative binoids

Proof. Obviously, ( (M), , spec M, ) is a commutative boolean binoid. Thus, the canonical binoid B ∩ ∅ epimorphism β : M (M), f D(f), factors through β¯ : M (M), [f] D(f), by → B 7→ bool → B 7→ Proposition 4.2.2. Clearly, β¯ is surjective and the injectivity follows from Lemma 4.2.3.

Example 4.2.6. By Example 2.2.21, we have spec(Nn)∞ = e I 1,...,n . Thus, {h iii∈I | ⊆{ }} D(f)= spec(Nn)∞ f = e supp f I = , {P ∈ | 6∈ P} {h iii∈I | ∩ ∅} which yields ((Nn)∞) = P . B ∼ n Observe that the canonical binoid epimorphism β : M (M) with f D(f), fulﬁlls (like the → B 7→ projection πbool)

ker β = f M D(f)= = f M f nilpotent = nil(M) , { ∈ | ∅} { ∈ | } where the equality in the middle is due to Lemma 4.1.5(5). In what follows, we will not distinguish between the booleanization (M , π ) and its topological realization ( (M),β). bool bool B Corollary 4.2.7. The canonical binoid epimorphism β : M (M), f D(f), induces a continuous →B 7→ semigroup isomorphism β∗ : spec (M) ∼ spec M. B −→ Proof. By Corollay 4.1.19, β∗ is a continuous embedding on im β∗. To prove that im β∗ = spec M let spec M and deﬁne := D(f) f . The fact that D(f) D(g) = D(f + g) shows that P ∈ Q { | ∈ P} ∩ Q is an ideal in (M), which is even prime since D(f) D(g) implies that f or g , hence B ∩ ∈ Q ∈ P ∈ P D(f) or D(g) . This shows that spec (M) with β∗( )= . ∈Q ∈Q Q ∈ B Q P Remark 4.2.8. The N-spectra of a binoid and of its booleanization are usually not isomorphic. For ∞ instance, we have N-spec N ∼= N by Lemma 1.5.7 but ∞ FC N-spec(N )bool = N-spec (x)/(2x = x) ∼= bool(N)

∞ by Example 1.5.5(1). In particular, if N = K is a ﬁeld of characteristic = 2, then K-spec(N )bool = ∞ 6 χ ,χ • = 0, but K-spec N = K. { {0} M } ∼ { ∞} ∼ Corollary 4.2.9. Every binoid homomorphism θ : M N induces a unique binoid homomorphism → (θ): (M) (N) such that the diagram B B →B

~~θ M / N~~

~~βM βN B(θ) (M) / (N) B B commutes. In particular, : comB boolB is a covariant functor from the category of commutative B → binoids into the category of commutative boolean binoids boolB.~~

~~Proof. By Proposition 4.2.2, the composition βN θ induces the desired unique binoid homomorphism between the booleanizations.~~

~~Now we relate the bidual of a binoid, which was studied in detail in Section 1.2, to its booleanization.~~

vv Corollary 4.2.10. Let M be a ﬁnitely generated binoid. The map (M) M with D(f) δf , v B →vv 7→ where δ (χ)= χ(f) for χ M , is a binoid isomorphism. In particular, M = M if and only if M f ∈ ∼ is boolean. 4.3 Dimension 119

Proof. The map (M) M vv of the statement is the binoid homomorphism δ¯ induced by the B → vv v canonical binoid homomorphism δ : M M , f δ , where δ : M 0, , δ (χ) = χ(f), cf. → 7→ f f →{ ∞} f Proposition 4.2.2 (see also Remark 1.5.13). In other words, there is a commutative diagram

δ vv M / M ✇; ✇✇ β ✇✇ ✇✇ ¯ ✇✇ δ (M) B ¯ ¯ v ∅ with δ(D(f)) = δf . To prove the bijectivity of δ, we use the identification M ∼= spec M via ∅ αP , spec M, cf. Corollary 4.1.2. For the injectivity let f,g M and suppose that ↔ P P ∈ ∅ ∈ δf (αP )= δg(αP ) (i.e. αP (f)= αP (g)) for all spec M. Thus, f if and only if g for every P ∈ ∈P vv ∈P spec M, which implies that D(f)=D(g). For the surjectivity let ψ M , ψ : spec∅ M 0, . P ∈ ∈ →{ ∞} Note that spec∅ M is a finite set by Proposition 2.2.20 because M is a finitely generated binoid. Since ψ is a binoid homomorphism, ker ψ is a superset-closed subset of spec∅ M. In particular, if A := spec∅ M ker ψ , then ¯ := is a prime ideal that does not lie in ker ψ. So {Q ∈ | Q 6∈ } Q Q∈A Q there is an element f for every ker ψ such that f ¯. Hence, ψ = δ = δ¯(D(f)) with P ∈ P P ∈ S P 6∈ Q f f := P∈ker ψ fP . CorollaryP 4.2.11. If M is a finitely generated binoid, then (M v )vv = M v .

~~Proof. If M is ﬁnitely generated, then so is M v by Proposition 2.2.20 and Corollary 4.1.2. Now the statement follows from Corollary 4.2.10~~

~~4.3 Dimension~~

Deﬁnition 4.3.1. The dimension of a nonzero binoid M, denoted by dim M, is deﬁned to be the combinatorial dimension of the spacespec M endowed with the Zariski topology, which is given by the supremum of the lengths of all chains of irreducible closed subsets in spec M. The dimension of the zero binoid is 1 by convention. − So far, we have seen that the ideal theory of a commutative binoid M resembles that of a ring, and the same is true for the topological theory of their spectra. Therefore, the following results will appear familiar again.

~~Lemma 4.3.2. Let M be a binoid. Then~~

dim M = sup ℓ , spec M . { |P0 ⊂P1 ⊂···⊂Pℓ Pi ∈ } Proof. By Lemma 4.1.4(3) and Proposition 4.1.16, every chain of prime ideals in M P0 ⊂···⊂Pℓ gives rise to a chain of irreducible closed subsets V( ) V( ) in spec M and vice versa. This Pℓ ⊂···⊂ P0 shows the equality.

Example 4.3.3. (1) By Example 2.2.21(1)-(3), we have for n 1, ≥ n dim(Nn)∞ = n , dim(N∞)n =2n 1 , and dim : N∞ =1 . − i=1 [ (2) By Example 2.2.7, dim P = dim P = n 1. n,∩ n,∪ − 120 4 Topology of commutative binoids

~~Corollary 4.3.4. dim M = dim Mred for every nonzero binoid M.~~

~~Proof. This follows from Corollary 2.2.11.~~

Corollary 4.3.5. Let M be a subbinoid of N. If N is integral over M with respect to the inclusion ι : M ֒ N, then dim M = dim N. In particular, for an arbitrary subbinoid M of N, one has → N dim M = dim M .

~~Proof. This follows from Corollary 2.2.16.~~

~~Corollary 4.3.6. If a binoid M admits a generating set with n elements, then dim M n. ≤ Proof. This follows from Proposition 2.2.20.~~

~~Proposition 4.3.7. A binoid M is a binoid group if and only if it is integral and dim M =0~~

Proof. The only if part is immediate. On the other hand, if M is integral, then is a prime ideal, {∞} and since dim M = 0, it is the only prime ideal. Hence, = M = M M ×. {∞} + \ Definition 4.3.8. Let M = . The h e i g h t of a prime ideal spec M is the supremum of the 6 {∞} P ∈ lenghts of strictly increasing finite chains in spec M that end with . In other words, P ht := sup ℓ = , spec M . P { |P0 ⊂···⊂Pℓ P Pi ∈ } The dimension of is the supremum of the lenghts of strictly increasing finite chains in spec M P that start with . In other words, P dim := sup ℓ = , spec M . P { |P P0 ⊂···⊂Pℓ Pi ∈ } For dim M =: d< , let F denote the number of all prime ideals spec M of dimension i, i N. ∞ i P ∈ ∈ Then Fk = 0 for all k > d and the (d + 1)-tuple

~~F (M) := (F0(M), F1(M),...,Fd(M)) is called the F - vector of M.~~

Minimal primes have height 0 and ht M = dim M, which implies F # min M and F (M) = 1. + d ≤ 0 Lemma 4.3.9. Let be a prime ideal of the binoid M. Then P ht = dim M and dim = dim M/ . P P P P Proof. This follows from Corollary 2.2.17 and Corollary 2.2.10.

Remark 4.3.10. As pointed out by Anderson and Johnsen in [3], Krull’s principal ideal theorem, which states that in a noetherian ring R every prime ideal p that is minimal over the ideal (r1,...,rk) ( R has ht p k ([23, Theorem 10.2]), need not be true for binoids that satisfy the ascending chain ≤ condition (a.c.c.) on ideals, or equivalently, in which all ideals are ﬁnitely generated. The binoid

M = FC(x, y)/(x + y =2x) is ﬁnitely generated, hence fulﬁlls the a.c.c. on ideals (and on congruences, cf. Remark 1.6.6). The prime ideals of M are given by ( x ( x, y = M , {∞} h i h i + 4.4 K - points 121 which shows that M is minimal over y , but ht M = 2 > 1. However, in [3, Theorem 4.4] it is + h i + shown that Krull’s principal ideal theorem is true for binoids that satisfy the a.c.c. on ideals and the following weaker cancellation property

~~a + b = a + c = c = u + b for some u M × . 6 ∞ ⇒ ∈ For cancellative1 monoids that satisfy the a.c.c. on ideals, Krull’s principal ideal theorem has been proved by Kobsa, cf. [38, Satz 13.7].~~

~~4.4 K - points~~

~~Convention. In this section, K always denotes a ﬁeld.~~

As the title of this section indicates, we are going to pursue a topological approach to binoids via their K - spectra, which were introduced in Section 1.5 (for arbitrary binoids K).2 For this, we tacitly assume basic knowledge of the following geometric objects in commutative algebra: the spectrum of a ring R, Spec R := p R p prime (ring) ideal in R , { ⊆ | } which is a topological space equipped with the Zariski topology, and the K - spectrum of a commutative K - algebra A, K-Spec A = HomK- alg(A, K) , which is a topological subspace of Spec A via

~~K-Spec A Spec A −→ with ϕ ker ϕ. For a detailed treatment of the topological spaces Spec A and K-Spec A, we refer to 7→ [47, Chapter 2 and 3].~~

~~By Proposition 3.2.3, we have the following isomorphism~~

K-spec M ∼= K-spec K[M] , where a K - point ϕ : M K of M corresponds one-to-one to the K -algebra homomorphismϕ ˜ : → K[M] K with → ϕ˜(F )= r ϕ(a) K a ∈ aX∈M for F = r T a K[M]. In particular, a characteristic point α , spec M, corresponds to a∈M a ∈ P P ∈ P α˜ : r T a r . P a 7−→ a aX∈M a∈XM\P The above identiﬁcation allows for regarding K-spec M as a topological space, where the closed sets are given by the aﬃne algebraic sets

V (F ) = ϕ K-spec M ϕ˜(F )=0 K { ∈ | } 1 Cancellative in the sense of monoid theory, which means a + b = a + c implies b = c. 2 Note that arbitrary binoids are written additively, but a ﬁeld K is a binoid with respect to the multiplication. 122 4 Topology of commutative binoids for F K[M] and ∈ V ( F ) := V (F ) K { i}i∈I K i i\∈I for a family (Fi)i∈I of elements in K[M]. The open subsets are given by the complements

D ( F ) := K-spec K[M] V ( F ) , K { i}i∈I \ K { i}i∈I for F K[M], i I. i ∈ ∈ Lemma 4.4.1. Let M be a binoid. (1) V ( A )= V (A ) for a family A K[M], i I, of subsets. K i∈I i i∈I K i i ⊆ ∈ (2) V ( a )= V (a ) for a family a K[M], j J, of ideals. K S j∈J j T j∈J K j j ⊆ ∈ (3) V (F G) = V (F ) V (G) for F, G K[M]. K P K T ∪ K ∈ (4) V ((F, G)) = V (F ) V (G), where (F, G) is the ideal in K[M] generated by F, G K[M]. K K ∩ K ∈ (5) V (ab) = V (a b) = V (a) V (b) for ideals a, b K[M]. K K ∩ K ∪ K ⊆ (6) V (A) V (B) for subsets A, B K[M] with B A. K ⊆ K ⊆ ⊆ (7) V (1) = and V (0) = K- spec K[M]. K ∅ K Proof. All assertions are easily veriﬁed or follow from the corresponding statements on Spec K[M].

Proposition 4.4.2. For a binoid homomorphism ϕ : M N, the induced semigroup homomorphism → ϕ∗ : K- spec N K- spec M, ψ ψϕ, is continuous, namely, if a is an ideal in K[M], then → 7→ ∗ −1 ∗ −1 (ϕ ) (DK (a)) = DK (aK[N]) and (ϕ ) (VK (a)) = VK (aK[N]) , where aK[N] is the extended ideal φ(a)K[N] under the K - algebra homomorphism φ : K[M] K[N] • → with r T a r T ϕ(a), a M . a 7→ a ∈ Proof. By taking complements, it suﬃces to prove the statement for the closed sets. Moreover, we only need to consider ideals of the form a = (F ) for one F K[M]. For these ideals, we have ∗ ∈ a ϕ (V (F )) = ψ K-spec N ψ˜(ϕ(F )) = 0 , where ψ˜(G) := • r ψ(a) if G = • r T , K { ∈ | } a∈M a a∈M a which implies the statement. P P Proposition 4.4.3. Let M be a binoid. (1) If is an ideal in M, then I K- spec(M/ ) = V (K[ ]) . I ∼ K I (2) For f M, we have ∈ K- spec M = D (T f ) = ϕ K- spec M ϕ(f) =0 . f ∼ K ∼ { ∈ | 6 } Proof. Both statements follow from the general fact that for a K - algebra homomorphism φ : A A′ → the induced map φ∗ : K-Spec A′ K-Spec A, ψ φψ, is continuous so that for an ideal a in A ∗ −1 → ′ 7→′ ′ one has (φ ) (VK (a)) = VK (φ(a)A ), where φ(a)A is the extended ideal of a in A via φ, cf. [47, Proposition 2.B.15]. Applying this to the K - algebra homomorphisms

~~K[M] K[M/ ] = K[M]/K[ ] and K[M] K[M ] = K[M] nf N , −→ I ∼ I −→ f ∼ {T |n∈ } cf. Corollary 3.2.8(3)&(4), proves the proposition. 4.4 K - points 123~~

Remark 4.4.4. If x ,...,x generate the binoid M, the canonical binoid epimorphism (Nn)∞ M, 1 n → e x with i 1,...,n , induces a surjective K - algebra homomorphism i 7→ i ∈{ } K[X ,...,X ] K[M] 1 n −→ with X T xi , i 1,...,n . Hence, there is an embedding i 7→ ∈{ } K-spec M An(K) −→ with ϕ (ϕ(x ),...,ϕ(x )), which shows that the topology on K-spec M can also be given as the 7→ 1 n subspace topology of An(K) (this is also true for the natural topology if K = R or C). In particular,

FC n K-spec n ∼= A (K) . ,[Under the above embedding K-spec M ֒ An(K), the subset V ((F ) ) K-spec M, F K[M → K i i∈I ⊆ i ∈ i I, can be identiﬁed with ∈ a An(K) F (a) = 0 for all i I , { ∈ | i ∈ } and the characteristic points α : M 1, 0 (K, , 1, 0) for spec M, which are independent of P →{ }⊆ · P ∈ K, are given by the 0-1-points

(α (x ),...,α (x )) An(K) . P 1 P n ∈ n ′ For instance, if M is positive, then αM+ corresponds to (0,..., 0) A (K). Moreover, one has ′ ∈ P⊆P for two prime ideals and if and only if α (x ) α ′ (x ) for all i 1,...,n . P P P i ≥ P i ∈{ } Example 4.4.5. The pictures below visualize the R - spectra of three diﬀerent commutative binoids in A2(R). The marked points are the characteristic points (spec M as a set) sitting inside the R - spectra.

~~R- spec FC(x,y)/(x + y = 0) R- spec FC(x,y)/(x + y = 3y) R- spec FC(x,y)/(2(x + y)=4(x + y))~~

Note that the C - spectrum of the binoid FC(x, y)/(x + y = 0) is irreducible in the Zariski and in the complex topology, whereas the R -spectrum is not even connected in the real topology as shown in the example above. This is a good example for the following result. Lemma 4.4.6. Let M be a torsion-free regular ﬁnitely generated binoid and K algebraically closed. Then K- spec M is irreducible. Proof. By Theorem 3.2.10, K[M] is a domain, so (0) Spec K[M] is the only minimal prime ideal. ∈ Hence, Spec K[M] is irreducible (this is true for all ﬁelds) by a result on Spec K[M] analogous to Corollary 4.1.17. See for instance [47, Corollary 3.A.14]. Since K is algebraically closed, K-spec M ∼= K-Spec K[M] coincides with the set of all maximal ideals of K[M], denoted by Spm K[M], cf. [47, Theorem 2.A.2]. This implies that Spm K[M] is a dense open subset of Spec K[M], cf. [47, Theorem 2.B.12 and Exercise 3.A.20]. Therefore, Spm K[M]= K-spec M is irreducible as well. 124 4 Topology of commutative binoids

By Remark 2.2.23, the semigroup N-spec M decomposes into subsemigroups that are monoids. This decomposition transfers to the topological situation (if N is a ﬁeld). Proposition 4.4.7. Let M be a binoid. Then

K- spec M = K- spec(M/ ) , P can P∈spec[ M where K- spec(M/ ) are closed subsets of K- spec M. In particular, if M is cancellative, then P can K- spec M = V (K[ ]) . K P P∈min[ M Proof. On the one hand, we have for every spec M a closed embedding P ∈ K-spec(M/ ) K-spec M P can −→ induced by the surjection π : M (M/ ) , cf. Proposition 1.5.6. On the other hand, every P → P can K - point ϕ : M K of M factors through (M/ ker ϕ) by Proposition 1.7.14(2), which proves → can K-spec M = K-spec(M/ ) . P can P∈spec[ M The supplement follows from Proposition 4.4.3(1) and the fact that for , spec M implies P⊆Q P Q ∈ that K[ ] K[ ], and hence V (K[ ]) V (K[ ]) by Lemma 4.4.1(6). P ⊆ Q K Q ⊆ K P Definition 4.4.8. Let M be a binoid. We call K-spec(M/ ) , spec M, the cancellative P can P ∈ K - p a r t s , and those that are maximal with respect to set inclusion are called the cancellative K - components. Cancellative components need not be irreducible though the name suggests so, cf. Example 4.5.4. Definition 4.4.9. Let M be a binoid, spec M, and ϕ := ιπ the canonical binoid homomorphism P ∈ P M π M/ ι diff(M/ ) , −→ P −→ P where π is the canonical projection onto the integral binoid M/ and ι the canonical binoid homo- P morphism to the difference group (M/ ) • = . With this notation, two elements f,g M P (M/P) 6 {∞} ∈ are called functionally equivalent if ϕ (f) = ϕ (g) for every spec M. The relation P P P ∈ ∼fe on M given by f g if f and g are functionally equivalent defines a congruence on M. ∼fe The name stems from the characterization of functionally equivalent elements in Proposition 4.4.11(3) below. Example 4.4.10. Nilpotent elements are functionally equivalent to because nf = implies that ∞ ∞ nϕ (f)= ϕ (nf)= for every spec M. Hence, ϕ (f)= since diff(M/ ) is a binoid group. P P ∞ P ∈ P ∞ P In particular, the class [ ] in M/ consists of all nilpotent elements by Corollary 2.3.10. ∞ ∼fe Proposition 4.4.11. Let M be a finitely generated binoid and f,g M. The following statements ∈ are equivalent. (1) f and g are functionally equivalent. (2) For every commutative binoid group G and every G - point ϕ : M G, one has ϕ(f)= ϕ(g). → (3) For every field K and every K - point ϕ : M K, one has ϕ(f)= ϕ(g). → (4) For every (some) algebraically closed field K with char K = 0 and every K - point ϕ : M K, → one has ϕ(f)= ϕ(g). 4.4 K - points 125

Proof. The implications (2) (3) (4) are trivial. For (1) (2) assume that f and g are function- ⇒ ⇒ ⇒ ally equivalent and that ϕ : M G is a binoid homomorphism to a commutative binoid group G. → By Lemma 1.7.3, ϕ factors through M/ ker ϕ, where ker ϕ =: is a prime ideal because G is a binoid P group. In particular, we have the following diagram

M ❏ ❏❏❏ ϕ ❏❏π❏ ❏❏❏ $ ϕ˜ M/ '/ G, ϕP P ι diﬀ(M/ ) P where kerϕ ˜ = . By Corollary 1.13.12, there is a unique binoid homomorphism φ : diﬀ(M/ ) G {∞} P → with φι =ϕ ˜, hence

~~ϕ(f)=ϕ ˜(π(f)) = φ(ιπ(f)) = φ(ϕP (f)) = φ(ϕP (g)) = φ(ιπ(g)) =ϕ ˜(π(g)) = ϕ(g) .~~

For (4) (1) take an algebraic closed field K of characteristic 0. Assuming that f and g are not ⇒ functionally equivalent, we find a prime ideal spec M with ϕ (f) = ϕ (g) in P ∈ P 6 P diff(M/ ) = Zr Z/pr1 Z Z/prs Z , P ∼ × 1 ×···× s where p is a prime number and r 1, i 1,...,s . Thus, for at least one k 1,...,r + s the kth i i ≥ ∈{ } ∈{ } entries of ϕ (f) and ϕ (g) in Zr Z/pr1 Z Z/prs Z, denoted by (ϕ ) (f) and (ϕ ) (g), do not P P × 1 ×···× s P k P k coincide. Since K is algebraically closed of characteristic 0, there are injective group homomorphisms

ι : Z K× and ι : Z/pri Z K× , 0 −→ i i −→ i 1,...,s . Hence, ι (ϕ ) is a K - point which separates f and g. ∈{ } k ◦ P k The condition on the characteristic of the field K in (4) of the preceding result is necessary because ∞ in G = (Z/pZ) , p prime, any two different elements are not functionally equivalent since idG is a G - point, but the only K - point of G into an (algebraically closed) field K of characteristic p is χG• .

Remark 4.4.12. In monoid theory, a monoid homomorphism ϕ : M K is called a K - character → of M. In case K = C or K = S1 = z C z = 1 C, such a homomorphism is simply called { ∈ | | | } ⊆ a c h a r a c t e r , and the set of all characters is a well-studied object of interest, see for instance [30, Chapter IV.2] or [15, Chapter 5.5]. A fundamental result, cf. [15, Theorem 5.58], says that the characters of a commutative group G separates its elements; that is, for any two elements a,b G ∈ there is a character ϕ such that ϕ(a) = ϕ(b). Similarly, cf. [15, Theorem 5.59], the characters of a 6 commutative monoid M separates its elements if and only if M is separative;thatis,2a = a+b =2b for a,b M implies a = b. By [28, Theorem 9.13], a commutative monoid M is separative if and ∈ only if M is free of asymptotic torsion. In this context, the preceding proposition suits the general monoid theory since asymptotically equivalent elements of a binoid are functionally equivalent (i.e. ). Indeed, f g is equivalent to nf = ng and (n + 1)f = (n + 1)g for some n 1 by ∼fe ≤ ∼ae ∼ae ≥ Lemma 1.4.18. If ϕ : M K is a K - point, then either ϕ(f)= ϕ(g)=0or ϕ(f), ϕ(g) K× with → ∈ ϕ(f)n+1 ϕ(g)n+1 ϕ(f) = = = ϕ(g) . ϕ(f)n ϕ(g)n

~~Hence, f g by Proposition 4.4.11. ∼fe 126 4 Topology of commutative binoids~~

~~We close this section with another characterization of functionally equivalent elements.~~

~~Lemma 4.4.13. Two elements f,g M are functionally equivalent if and only if Xf Xg is nilpotent ∈ − in K[M] for every ﬁeld K.~~

Proof. Let F := Xf Xg be nilpotent. If ϕ : M K is a K - point, there is by Proposition − → 3.2.3 a unique K - algebra homomorphism Φ : K[M] K such that ϕ = Φι, where ι : M → → K[M] is the canonical binoid homomorphism a Xa, a M. Since F is nilpotent and K a 7→ ∈ field, 0 = Φ(F ) = Φ(Xf ) Φ(Xg), which gives ϕ(f) = Φ(Xf ) = Φ(Xg) = ϕ(g). Conversely, − if F is not nilpotent in K[M] for some field K, then F n n 0 is a multiplicative system in { | ≥ } K[M] such that K[M] = 0. In particular, Spec K[M] = . Hence, there is a ring homomorphism F 6 F 6 ∅ Φ: K[M] Q(K[M] /p)= L into the quotient field of K[M] /p, p Spec K[M] , with Φ(F ) = 0. F → F F ∈ F 6 This is equivalent to Φ(Xf ) = Φ(Xg), which implies that 6 ι ϕ : M ι K[M] F K[M] Φ L −→ −→ F −→ is an L - point of M with ϕ(f) = ϕ(g). 6

Any two different elements f,g (Z/pZ)∞, p prime, are not functionally equivalent as remarked ∈ after Proposition 4.4.11. However, X 1 = X1 X0 is nilpotent in K[(Z/pZ)∞] = K[X]/(Xp 1) − − ∼ − for every field K with characteristic p because (X 1)p = Xp 1. This shows that Xf Xg being − − − nilpotent needs to be checked for every field. In particular, the ring K[(Z/pZ)∞ is not reduced if char K = p, whereas (Z/pZ)∞ is a reduced binoid. As another example consider the (reduced) binoid

~~M := FC(a,b)/(2a = a + b =2b) .~~

Since 3a = a +2a = a +2b = (a + b)+ b = 2b + b = 3b the generators a and b are asymptotically equivalent, hence functionally equivalent by Remark 4.4.12. Thus, Xa Xb is a nilpotent element in K[M] for every ﬁeld K by the preceding lemma. −

~~4.5 Connectedness properties of K - spectra~~

~~Convention. In this section, K always denotes a ﬁeld.~~

Recall that a nonempty topological space is said to be c o n n e c t e d if it is not the union of two disjoint nonempty open (resp. closed) sets, otherwise the space is said to be d i s c o n n e c t e d . We want to know which assumptions on a binoid M ensure connectedness of K-spec M. Of course, every idempotent e in M yields an idempotent in K[M], namely Xe. Then, K-spec M is not connected since it is the disjoint union of the closed sets V (Xe)= ϕ K-spec M ϕ(e)=0 and V (Xe K { ∈ | } K − 1) = ϕ K-spec M ϕ(e)=1 , which are not empty since K-spec M/ e and K-spec M are { ∈ | } h i e not empty (this follows from the fact that both binoids, M/ e and M , are not empty, and hence h i e = spec K- spec). On the other hand, idempotents in K[M] do not need to come from idempotents ∅ 6 ⊆ in M as the subsequent example shows. At the end of this section we will show that in case of hypersurfaces, the non-existence of non-trivial combinatorial idempotents is equivalent to connectedness in the torsion-free situation, cf. Corollary 4.5.8. First we state a criterion for connectedness and study hypersurfaces in general. 4.5 Connectedness properties of K - spectra 127

Example 4.5.1. The binoid (Z/2Z)∞ is not torsion-free and contains no idempotent elements, whereas its K - algebra K[(Z/2Z)∞]= K[X]/(X2 1) contains the idempotent 1 (X +1) if char K =2 − 2 6 since X +1 2 X2 +2X +1 1+2X +1 X +1 = = = . 2 4 4 2 In particular, if char K = 2, then K-spec(Z/2Z)∞ is not connected. 6 The following result is our main combinatorial criterion for connectedness.

Theorem 4.5.2. Let M be a ﬁnitely generated torsion-free binoid and K algebraically closed. The following statements are equivalent: (1) K- spec M is connected. (2) For any two cancellative K - components X and Y of M, there is a sequence X(1),...,X(n) of cancellative K - components with X(1) = X and X(n)= Y such that X(i) X(i + 1) contains a ∩ K - point, i 1,...,n 1 . ∈{ − } (3) For any two cancellative K - components X and Y of M, there is a sequence X(1),...,X(n) of cancellative K - components with X(1) = X and X(n)= Y such that X(i) X(i + 1) contains a ∩ characteristic point, i 1,...,n 1 . ∈{ − } Proof. The equivalence (1) (2) follows from the deﬁnition of connectedness and from Proposition ⇔ 4.4.7 since all cancellative K - components of M are irreducible closed subsets of K-spec M by Lemma 4.4.6. The implication (3) (2) is trivial. ⇒ To prove (2) (3), it is enough to show that any two cancellative K - components, say X = ⇒ K-spec(M/ ) and Y = K-spec(M/ ) for , spec M, with a nonempty intersection contain P can Q can P Q ∈ a common characteristic point of M. Assuming X Y = , we have a K - point of M that factors ∩ 6 ∅ through (M/ ) and through (M/ ) via M - binoid homomorphisms P can Q can

(M/ )can ✉: P ❏❏ πP ✉✉ ❏❏ϕP ✉✉ ❏❏ ✉✉ ❏❏ ✉✉ ❏% M ❏ K. ❏❏ tt: ❏❏ tt ❏❏ tt πQ ❏ ttϕQ ❏$ tt (M/ ) Q can By Corollary 1.10.22, there is an M - binoid homomorphism

(M/ ) (M/ ) K, P can ∧M Q can −→ in particular, (M/ ) (M/ ) is not the zero binoid. Thus, there is a characteristic point P can ∧M Q can α : (M/ ) (M/ ) 1, 0 K, and therefore P can ∧M Q can →{ }⊆

(M/ )can ♠♠6 P πP ♠♠♠ ♠♠♠ ιP ♠♠♠ ♠♠♠ α M ◗◗ / (M/ )can M (M/ )can / 1, 0 , ◗◗◗ P ∧ O Q { } ◗◗◗ ◗◗ ιQ πQ ◗◗◗ ◗( (M/ ) Q can where αι π = αι π : M K is a common characteristic point of M on X and Y . P P Q Q → 128 4 Topology of commutative binoids

~~Example 4.5.3. The binoid M := FC(x, y)/(x + y = y) satisﬁes all assumptions of Theorem 4.5.2. Its R - spectrum~~

~~R- spec FC(x,y)/(x + y = y) is connected and the two cancellative R - components meet in a characteristic point. To see this consider spec M, which consists of three prime ideals, namely~~

~~( y ( x, y = x . {∞} h i h i h i These yield the three cancellative parts:~~

∞ R-spec Mcan ∼= R-spec N ∼= R , which corresponds to the cancellative R - component given by the vertical line since x 0, cf. ∼can Lemma 1.5.7, R-spec(M/ y ) = R-spec N∞ = R , h i can ∼ ∼ which corresponds to the cancellative R - component given by the horizontal line, and

~~R-spec(M/ x, y ) = R-spec 0, = , h i can { ∞} ∼ {∞} which corresponds to the special point, cf. Example 1.5.5(2).~~

~~Example 4.5.4. If K contains the 8th roots of unity ζ0,...,ζ7, for instance if K = C, one may imagine the K - spectrum of the binoid M := FC(x, y)/(8x + y = y) in the following way:~~

~~K- spec FC(x,y)/(8x + y = y)~~

The binoid satisﬁes all assumptions of Theorem 4.5.2 except the torsion-freeness since 8y = (8x + y)+ 7y = 8(x+y). However, its K - spectra is connected and all cancellative K - components are connected by characteristic points. To see this consider spec M, which is given by the three prime ideals

~~( y ( x, y = x . {∞} h i h i h i 4.5 Connectedness properties of K - spectra 129~~

~~These yield the three cancellative parts:~~

K-spec M = K- spec((Z/8Z)∞ N∞) = ζ ,...,ζ K, can ∼ ∧ ∼ { 0 7}× which corresponds to the cancellative K - component given by the 8 vertical lines since 8x 0, cf. ∼can Lemma 1.12.3 and Proposition 1.8.12,

K-spec(M/ y ) = K-spec N∞ = K, h i can ∼ ∼ which corresponds to the cancellative K - component given by the horizontal line, cf. Lemma 1.5.7, and K-spec(M/ x ) = K-spec 0, = , h i can ∼ { ∞} ∼ {∞} which corresponds to the special point, cf. Example 1.5.5(2). Now we study the case of hypersurfaces. Proposition 4.5.5. Let K be a ﬁeld of characteristic zero and consider the binoid

M := FC(x1,...,xn)/(f = g) with f = f x + + f x and g = g x + + g x , so that 1 1 ··· n n 1 1 ··· n n K[M] = K[X]/(Xf Xg) , − where Xν := Xν1 Xνn for ν = (ν ,...,ν ) Nn. 1 ··· n 1 n ∈ (1) If g =0, so that K[M] = K[X]/(Xf 1) , − then K- spec M is connected if and only if M is torsion-free. In this case, there are r, s 0 such ≥ that K- spec M = Ar(K) (K×)s . × (2) If g =0 and f = f˜+ g, so that 6 ˜ K[M] = K[X]/Xg(Xf 1) , − then K- spec M is disconnected if supp g supp f˜ and connected otherwise. ⊆ (3) If f = f˜+ h and g =˜g + h such that supp f,˜ suppg ˜ = with supp f˜ suppg ˜ = , i.e. 6 ∅ ∩ ∅ ˜ K[M] = K[X]/Xh(Xf Xg˜) , − then K- spec M is connected. Proof. (1) In this situation, we have by the structure theorem for finitely generated commutative groups M = (Nr Zs Z/k Z Z/k Z)∞, where r := # i f =0 . Thus, K[M]= K[(Nr)∞] ∼ × × 1 ×···× l { | i } ⊗K K[(Zs)∞] K[T ∞], which gives ⊗K K-spec M = Ar(K) (K×)s K-spec T ∞ , × × where T := Z/k Z Z/k Z is a finite torsion group. In particular, M is not torsion-free if and 1 ×···× l only if T = 0 , which is equivalent to K-spec T ∞ is a finite set with more than one point. Therefore, 6 { } K-spec M is connected if and only if M is torsion-free. (2) First let supp g supp f˜ and choose k 1 such that kf˜i gi for all i supp f˜. By assumption, ˜ ⊆ ≥ ˜ ≥ ˜ ∈ ˜ ˜ we have g = f = f + g, which implies that g = kf + g, where kf = i∈supp f˜ kfixi. Hence, kf is an P 130 4 Topology of commutative binoids idempotent element in M because

2kf˜ = (kf˜+ g) + (kf˜ g)= g + (kf˜ g)= kf˜ , − − in particular Xkf˜ is idempotent in K[M], so K-spec M is not connected. Now let supp g supp f˜, 6⊆ say supp g = 1,...,r but 1 supp f˜. Then { } 6∈ ˜ ˜ K-spec M = V (Xg(Xf 1)) = V (Xg) V (Xf 1) , K − K ∪ K −

g r gi r n where VK (X )= i=1 VK (Xi )= i=1 VK (Xi) is connected because the origin (0,..., 0) A (K) gi f˜ ∈ lies in each VK (Xi ), i supp g. Moreover, though VK (X 1) might not be connected itself, every S ∈ S f˜ − 1 K - point a = (a1,a2,...,an) VK (X 1) lies on the line La := (t,a2,...,an) t A (K) f ∈ −f˜ { | ∈ g1 } ⊆ VK (X 1) since X1 does not occur in X , on which the K - point (0,a2,...,an) VK (X1 ) lies as − ˜ ∈ well. Therefore, every K - point of V (Xf 1) is connected with V (Xg). K − K (3) Here,

˜ ˜ K-spec M = V (Xh) V (Xf Xg˜) = V (Xhi ) V (Xf Xg˜) , K ∪ K − K i ∪ K − i∈supp[ h where V (Xf˜ Xg˜) is connected and contains the origin (0,..., 0), cf. Corollary 5.2.12, which also K − lies in every V (Xhi ), i supp h. K i ∈ Note that the assumption on K being algebraically closed of characteristic zero in the preceding proposition may make things worse, that is to say, in this case the K - spectrum of a binoid is more likely to be disconnected. For instance, in dimension 0 or in case of one generator, the K - spectrum V (Xn 1) of the (non-torsion-free) binoid (Z/nZ)∞ with n 2 such that n 1 mod 2 and K − ≥ ≡ char K = 0, consists only of a single point and is therefore connected if K does not contain the nth roots of unity. Otherwise, K-spec(Z/nZ)∞ is a set with n 2 points, hence hausdorﬀ, cf. Example ≥ r r 5.2.11. On the other hand, if char K = p> 0, then K-spec(Z/prZ)∞ = V (Xp 1)=V ((X 1)p ) K − K − is a singleton as well.

~~Example 4.5.6. We have~~

K-spec FC(x,y,z)/(x + y +2z = x + y) = V (XY (Z2 1)) = V (X) V (Y ) V (Z2 1) , K − K ∪ K ∪ K − and these planes are connected for K inﬁnite. In this case, the connected sets VK (X) and VK (Y ) intersect in the line L = (0, 0,a) a A1(K) . Though the closed set V (Z2 1) decomposes { | ∈ } K − further into the disjoint connected sets V (Z 1) and V (Z + 1) (i.e. V (Z2 1) is not connected), K − K K − K-spec FC(x,y,z)/(x + y +2z = x + y) is connected since L intersects with V (Z 1) in (0, 0, 1) and K − with V (Z + 1) in (0, 0, 1). K −

~~R- spec FC(x,y,z)/(x + y + 2z = x + y) 4.5 Connectedness properties of K - spectra 131~~

~~Example 4.5.7. For K inﬁnite, we have~~

K-spec FC(x,y,z)/(2x + y + z = x + z) = V (XZ(XY 1)) K − = V (X) V (Z) V (XY 1) . K ∪ K ∪ K − The connected sets V (X) and V (Z) intersect in the line L = (0, t, 0) t A1(K) but V (X) K K { | ∈ } K and V (XY 1) = K× are disjoint. However, every K - point (a,b,c) V (XY 1) lies on the line K − ∼ ∈ K − L := (a,b,t) t A1(K) which intersects V (Z) in (a, b, 0). ab { | ∈ } K

~~R- spec FC(x,y,z)/(2x + y + z = x + z)~~

Corollary 4.5.8. Let M be a torsion-free binoid such that M ∼= FC(x1,...,xn)/(f = g). If K is a ﬁeld of characteristic zero, then K- spec M is connected if and only if M contains no non-trivial idempotent elements.

Proof. In general (i.e. regardless if M is torsion-free), if e M is a non-trivial idempotent element, ∈ then Xe is idempotent in K[M], so K-spec M is not connected. On the other hand, K-spec M is disconnected if and only if e =e ˜ + g and supp g suppe ˜ by Proposition 4.5.5. In this case, the ⊆ element ke˜, where k 1 satisﬁes kf˜ g for all i 1,...,n , is an idempotent element of M, cf. ≥ i ≥ i ∈ { } the proof of 4.5.5(2).

Example 4.5.9. Let K be an algebraically closed ﬁeld. For any two elements f,g FC(x ,...,x ) ∈ 1 n with i supp f and j supp g, but j supp f and i supp g for some i, j 1,...,n , the binoids ∈ ∈ 6∈ 6∈ ∈{ }

~~FC(x1,...,xn)/(mf = mg) , m 2, are not torsion-free, but their K -spectra are connected if char K = 0 by Proposition 4.5.5(3). ≥~~

~~5 Separation and gradings~~

This chapter is concerned with the question when the special point is contained in every cancellative component. For this, we ﬁrst study the separating ideal n≥1 nM+ of a commutative binoid. We are interested in the case when this ideal is , in which we call M a separated binoid. Under certain {∞} T conditions the separating ideal can easily be described, cf. Proposition 5.1.6, which yields the notion of (un-) separated elements. We show that the existence of an unseparated element gives a necessary condition to our question, cf. Proposition 5.1.17. The second part of this chapter deals with gradings on a binoid (by a monoid). We are particularly interested in positive Nk -gradings, whose existence is equivalent to M being separated under certain assumptions, cf. Theorem 5.2.7, and which yield a positive result to our initial question, cf. Theorem 5.2.10.

~~5.1 Separated binoids~~

~~Convention. In this section, arbitrary binoids are assumed to be commutative.~~

The main focus of this section is the ideal n≥1 nM+. As in ring theory, it is of interest to know when this is the zero ideal because this has topological consequences regarding the K -spectra. For T a treatment of this ideal within commutative algebra of binoids, the reader may consult [3], and [38] for that of monoids.

∞ Deﬁnition 5.1.1. The ideal n=1 nM+ is called the s e p a r a t i n g ideal of M. A nonzero binoid M is called s e p a r a t e d if T nM = . + {∞} n\≥1 ∞ The separated dimension of a binoid is deﬁned to be dim(M/ n=1 nM+) and will be denoted by dim M. sep T

~~If M is separated, then dim M = dimsep M, but the converse need not be true, cf. Example 5.1.7.~~

~~∞ ∞ ∞ Example 5.1.2. The binoid N is separated because n(N )+ = N≥n, and hence~~

n(N∞) = (N∞ ) = . + ≥n + {∞} n\≥1 n\≥1 Remark 5.1.3. The definition of being separated stems from the terminology in ring theory, which is based on the fact that the a - adic topology defined on a commutative ring R, a R an ideal, is ⊆ separated (hausdorff) if and only if an = 0 , { } n\≥1 cf. [4, Chapter 10] or [31, Chapter 0 7.2]. Recall that the a - adic completion of R is defined to be the § projective limit of the inverse system (R/an, φ : R/am R/an) and is denoted by nm → m≥n≥0 lim R/an := R.ˆ ←− 134 5 Separation and gradings

~~Similarly, every binoid M deﬁnes an inverse system (Mn, ϕnm)m≥n≥0 of binoids, where~~

~~M := M/nM and ϕ : M M , ,m n , n + nm m → n ≥ are the canonical projections. The projective limit~~

lim M = (a ) M ϕ (a )= a ,m n , n n n≥0 ∈ n mn n m ≥ ←− n n≥0 o Y cf. Lemma 1.11.6, may be considered as the completion of M. The inverse system (Mn, ϕnm)m≥n≥0 can be illustrated as follows

ϕk,k+1 ϕ ϕ M/(k + 1)M M/kM M/2M 12 M/M 01 0, . · · · −→ + −→ + −→ · · · −→ + −→ + −→ { ∞} Since the binoid homomorphisms ϕ , m n, are injective on M nM , every = (a ) nm ≥ m \ + ∞ 6 n n≥0 ∈ lim Mn is of the form ←− ( ,..., , a, a, a, . . .) ∞ ∞ (by abuse of notation we write a for the class [a]n in Mn if a nM+). More precisely, an = for •6∈ ∞ 0 n k and a = a M for n k +1 and some a M , where k 0 is the index such that ≤ ≤ n ∈ n ≥ ∈ ≥ a kM but a (k + 1)M . In particular, if M is separated, there is a one-to-one correspondence ∈ + 6∈ + a ([a] ) , which yields ↔ n n≥0 M ∼= lim Mn . ←− If, in addition, M is positive, then K[M+] =: m is a maximal ideal in K[M] by Proposition 3.3.2. Therefore, n \ lim K[Mn] = lim K[M]/m = K[M] . ←− ←− In particular, if K[M] is not complete (i.e. K[M] = K\[M]), then K[lim M ] = lim K[M ]. 6 n 6 n ←− ←− Lemma 5.1.4. Let M be a nonzero binoid.

(1) M/ n≥1 nM+ is separated. (2) If M is integral with dim M = dim M < , then M is separated. T sep ∞ (3) If M is positive and separated, then so is every subbinoid of M. (4) If M is separated, then so is M/ for every ideal M. I I ⊆ Proof. (1) Set := nM . By Corollary 2.2.10, (M/ ) = M / which implies the assertion. I n≥1 + I + + I (2) If ∞ nM = , then the prime ideal is contained in ∞ nM , and hence dim M < n=1 + 6 {∞}T {∞} n=1 + sep dim M by Corollary 2.2.10, contrary to our assumption. (3) Let N M be a subbinoid and ι : N ֒ M T ⊆T → the canonical binoid embedding. We have ι(N ) M by the positivity of M, which implies that + ⊆ + ι( nN ) nM = . Thus, N is separated as well (the positivity is trivial). (4) n≥1 + ⊆ n≥1 + {∞} Suppose that [f] = [ ] for some [f] n(M / ). For every n 1, we ﬁnd a ,...,a (M ) T 6T ∞ ∈ n≥1 + I ≥ 1 n ∈ + \ I such that f = a + + a = in M. This means = f nM for every n 1, a contradiction 1 ··· n 6 ∞ T ∞ 6 ∈ + ≥ to M separated.

~~Lemma 5.1.5. Let (Mi)i∈I be a ﬁnite family of nonzero binoids.~~

(1) If all Mi are separated, then i∈I Mi is also separated. Conversely, if i∈I Mi is positive and separated, then all M are so. i V V : (2) If all Mi are positive, then i∈I Mi is separated if and only if all Mi are so. S Proof. (1) Set M := i∈I Mi. First let all Mi be separated and assume that f nM+ for every • ∈ n 1, where = f = f M, f M . Since M is separated, there is for every i I an ≥ ∞∧ 6 V ∧i∈I i ∈ i ∈ i i ∈ 5.1 Separated binoids 135 m 1 with f l(M ) for all l m . Let m := m and I = 1,...,r . By the description of i ≥ i 6∈ i + ≥ i i∈I i { } the prime ideals in the smash product, cf. Corollary 2.2.12, we have P mM = m M (M ) M . + 1 ∧···∧ k + ∧···∧ r k[∈I Thus, for every g = g mM there is at least one i I such that g lM for some l m , ∧i∈I i ∈ + ∈ i ∈ i ≥ i which means f cannot be contained in mM+, a contradiction. For the converse let M be positive and separated. Each Mk can be considered as a subbinoid of M via the canonical binoid embedding ι : M ֒ M, a ae . Hence, all M are separated and positive by Lemma 5.1.4(3) and Lemma k → 7→ k k 1.8.8(2). The second statement is easily veriﬁed. b Proposition 5.1.6. Let M be a binoid. Then

~~f M f = f + g for some g M nM , { ∈ | ∈ +} ⊆ + n\≥1 and equality holds if M is ﬁnitely generated. In particular, if M is positive and ﬁnitely generated, then~~

nM = f M f = f + g for some g =0 . + { ∈ | 6 } n\≥1 Proof. If f = f +g for some g M , then in particular f = f +g M , which gives f = f +ng nM ∈ + ∈ + ∈ + for all n 1. This proves the inclusion. For the equality, we may assume that x ,...,x is a ≥ { 1 l} generating set of M with x ,...,x M and x ,...,x M ×. If f nM , there is for 1 k ∈ + k+1 l ∈ ∈ n≥1 + every n 1 at least one k - tuple r = (r ,...,r ) Nk such that f = r x + + r x + u with ≥ 1 k ∈ 1 1 T ··· k k r u M × and r + + r n. In particular, the set r ∈ 1 ··· k ≥ K = (r ,...,r ) Nk f = r x + + r x + u for some u M × { 1 k ∈ | 1 1 ··· k k r r ∈ } is infinite. By Dicksons’s Lemma, cf. [48, Theorem 5.1], the subset K′ K of minimal elements in ⊆ K with respect to the product order is finite, in particular K′ ( K. By the definition of a minimal element and an easy finiteness argument, see [48, Corollary 5.4], there is for every r K an s K′ ∈ ∈ with s r. Thus, we find a pair (s, r) K′ K with s = r and s r, which yields ≤ ∈ × 6 ≤ f = r x + + r x + u = s x + + s x + u 1 1 ··· k k r 1 1 ··· k k s for some u ,u M ×. This shows that f = f + g with g = u + (-u )+ k (r s )x M . The r s ∈ r s i=1 i − i i ∈ + supplement is clear. P Example 5.1.7. The prime ideal lattice of the binoid M = FC(x,y,z)/(x + y = x + y + z, 2z = ) ∞ is given by x,y,z h i ⊃ ⊂ x, z y,z h i h i ⊃ ⊂ z h i By Proposition 5.1.6, the separating ideal of M is x + y . Since there is no spec M with h i P ∈ x + y , the prime ideal lattices of M and M/ x + y are one-to-one by Corollary 2.2.10. In P ⊆ h i h i particular, dim M = dimsep M = 2. 136 5 Separation and gradings

Consider now the integral binoid N = FC(x,y,z)/(x + y = x + y + z) (i.e. omit the relation 2z = ), ∞ whose separating ideal is x + y by Proposition 5.1.6, and whose prime spectrum is spec N = J h i {h i | J P( x,y,z ) . In particular, is a prime ideal contained in x + y . This gives diﬀerent ∈ { } } {∞} h i dimensions, namely dim N = 3 and dimsep N = 2.

Example 5.1.8. Here is an example, where dim M, dimsep M, and the dimension Dim R[M] of the ring R[M] are pairwise diﬀerent. The prime ideal lattice of the binoid M = FC(x,y,z)/(y + x = y,z + x = z) is given by x,y,z = x h i h i ⊃

~~y,z h i ⊃ ⊂ y z h i h i ⊃ ⊂~~

{∞} Hence, dim M = 3. By Proposition 5.1.6, the separating ideal is y,z , so dim M = 1. Note that h i sep for a field K the Krull dimension Dim K[M] = Dim K[X,Y,Z]/(YX Y,ZX Z) is 2. − − Proposition 5.1.6 gives rise to the following definition. Definition 5.1.9. An element f M is called w e a k l y s e p a r a te d if f = or f = f + g for some ∈ ∞ g M implies g M ×, otherwise f is weakly unseparated. We say f M is s e p a r a t e d if ∈ ∈ ∈ f = or f = f + g for some g M implies g = 0, otherwise f is u n s e p a r a t e d . ∞ ∈ Of course, separated implies weakly separated and the converse holds true if M is positive. When dealing with positive binoids, we will not always mention this (and omit “weakly”). Cancellative elements are always separated and so are the absorbing and the identity element of a binoid, whereas idempotent elements 0, are not even weakly separated. A binoid in which all elements = 0 are 6∈ { ∞} 6 nilpotent contains only the trivial separated elements since x = x + y for some y = 0 implies that 6 ny = for some n 2, and hence x = x + y = x +2y = = x + ny = . ∞ ≥ ··· ∞ Corollary 5.1.10. In a separated binoid M all elements are weakly separated, and the converse holds true if M is finitely generated. In particular, a positive and finitely generated binoid is separated if and only if all elements are separated. Proof. This is an immediate consequence of Proposition 5.1.6.

Example 5.1.11. Consider the positive binoid (Q∞ , +, 0, ), which is not ﬁnitely generated, cf. ≥0 ∞ Example 1.3.13(5). We have (Q∞ ) = Q∞ and a/b = n(a/nb) nQ∞ for every a/b Q∞ and ≥0 + >0 ∈ >0 ∈ >0 n 1. This shows that Q∞ is not separated because ≥ ≥0

∞ ∞ n(Q>0)+ = Q>0 , n\≥1 ∞ but all elements are separated since Q≥0 is cancellative. Lemma 5.1.12. Let M be a nonzero binoid and ( M an ideal. If f M is (weakly) separated, I ∈ then so is [f] M/ . ∈ I Proof. This is immediate since [f] = [f] + [g] in M/ is equivalent to f = f + g or f , and since I ∈ I = M implies that M × = , we have (M/ )× = [u] u M × . I 6 I ∩ ∅ I { | ∈ } 5.1 Separated binoids 137

Lemma 5.1.13. Let (Mi)i∈I be a finite family of nonzero binoids. (1) An element f M is (weakly) separated if and only if all f M are so. ∧i∈I i ∈ i∈I i i ∈ i (2) If all M are positive, then (f; k) : M is separated if and only if f M is so. i V ∈ i∈I i ∈ k Proof. (1) If f = or (equivalently)Sf = for some i I, the statement is trivial. So assume ∧i∈I i ∞∧ i ∞ ∈ that all f = are (weakly) separated and i 6 ∞ f = ( f ) + ( g )= (f + g ) ∧i∈I i ∧i∈I i ∧i∈I i ∧i∈I i i for some g M , i I. By the definition of the smash product, = f = f + g for all i I i ∈ i ∈ ∞ 6 i i i ∈ which implies that g = 0 (g M ×) by assumption, and hence g = 0 ( g ( M )×, i i ∈ ∧i∈I i ∧ ∧i∈I i ∈ i∈I i cf. Lemma 1.8.8(2)). Conversely, if = f is (weakly) separated and f = f + g for some ∞∧ 6 ∧i∈I i k k Vk k I and g M , then f = and = f = f + g e . This implies that g e = 0 ∈ k ∈ k k 6 ∞ ∞∧ 6 ∧i∈I i ∧i∈I i k k k k ∧ (g e ( M )×) by assumption, hence g = 0 (g M ×). (2) is trivial. k k ∈ i∈I i k k ∈ k b b LemmaV 5.1.14. Let M be a binoid consisting only of (weakly) separated elements and S M a b ⊆ submonoid with S. ∞ 6∈ (1) If M is positive, then so is Mcan,S. (2) If S int(M), then M consists only of (weakly) separated elements as well. ⊆ can,S Proof. (1) Note that in a positive binoid the notions of weakly separated and separated are equivalent. To show that M is positive suppose that [0] = [f] + [g] = [f + g] in M for f,g M. This can,S can,S ∈ is equivalent to f + g + s = s in M for some s S. By assumption, s = or f + g = 0, and since ∈ ∞ S the latter holds. Thus, f = g = 0 by the positivity of M, hence [f] = [g] = [0]. (2) Assume ∞ 6∈ that [f] = [f] + [g] = [f + g] in M . Then f + s = g + f + s in M for some s S, which implies can,S ∈ that g = 0 (g M ×) or f + s = by assumption. Since s is an integral element, we obtain g = 0 ∈ ∞ (g M ×) or f = . Thus, [g] = 0 ([g] (M )×) or [f] = [ ]. ∈ ∞ ∈ can,S ∞ Definition 5.1.15. An order function onabinoid M is a map ord : M • N such that → ord(f + g) ord(f) + ord(g) ≥ for all f,g M with f + g = . Such a function is called p o s i t i v e if ord(f) > 0 for all f M . ∈ 6 ∞ ∈ + Since there are no idempotent elements = 0 in N, every order function satisfies ord(0) = 0, which 6 implies ord(u) = 0 for all u M ×. ∈ Proposition 5.1.16. If M is separated, then

• δ : M N , f max k N f kM , −→ 7−→ { ∈ | ∈ +} where 0M+ := M, is a positive order function on M. Conversely, a binoid is separated if it is ﬁnitely generated and admits a positive order function.

Proof. It is easily checked that δ deﬁnes a positive order function because the convention 0M+ = M implies that δ(f) = 0 if and only if f is a unit. For the converse let ord : M • N be a positive order → function. Since M is ﬁnitely generated, we have

f M f = f + g for some g M = nM { ∈ | ∈ +} + n\≥1 by Proposition 5.1.6. Therefore, we only need to show that is the only weakly unseparated element. • ∞ For this assume that f = f + g for some f,g M . Then ord(f) ord(f)+ ord(g) in N which implies ∈ ≥ that ord(g) = 0, and hence g M ×. ∈ 138 5 Separation and gradings

Proposition 5.1.17. Let M be a ﬁnitely generated reduced binoid and K a ﬁeld. If M contains an unseparated element, then there exists an irreducible component of K- spec M that does not contain the special point.

Proof. By assumption, there are elements f,g M with = f = f + g and g = 0. Since M is ∈ ∞ 6 6 reduced, the element f + g is not nilpotent, which implies that D(f + g) = by Lemma 4.1.5(5). Take 6 ∅ any D(f + g) and let Q := α : M K be the (characteristic) K -point corresponding to , P ∈ P → P which satisﬁes α (f + g) = α (f) = 1 and α (g) =1. Let S := α : M K denote the special P P P M+ → point. We have T f = T f T g in K[M], and therefore V (T f ) V (T g 1) = K-spec M by Lemma K ∪ K − 4.4.1(3)&(7). By the above considerations, the K - point Q lies in V (T g 1) but is not contained K − in V (T f ). In particular, there is an irreducible component C V (T g 1) with Q C. Then K ⊆ K − ∈ for every K - point ϕ C, one has ϕ(T g) = 1, and therefore S C since g M . Thus, C is the ∈ 6∈ ∈ + irreducible component in question.

~~Example 5.1.18. Consider the following R - spectra:~~

~~R- spec FC(x,y)/(x + 2y = 2x + y) R- spec FC(x,y)/(x + y = x) R- spec FC(x,y)/(x + y = x, 2x = ∞)~~

The binoid that yields the R - spectrum on the left-hand side is separated, whereas the other binoids are not. These two only diﬀer in being reduced or not and show that this condition is necessary in the preceding proposition.

Corollary 5.1.19. Let M be a ﬁnitely generated binoid that is positive and reduced. If M is not separated, then there exists an irreducible component of K- spec M that does not contain the special point.

~~Proof. This is an immediate consequence of Proposition 5.1.17 and Corollary 5.1.10.~~

~~5.2 Graded binoids~~

~~In this section, we consider binoids that admit a grading and study their K -spectra for a ﬁeld K. Graded binoids also appear in [19].~~

Deﬁnition 5.2.1. Let M be a binoid and D a monoid. We call M a D - g r a d e d binoid if there is a map δ : M • D such that δ(0) = 0 and for all x, y M • : → ∈ δ(x + y) = δ(x)+ δ(y) when x + y = . 6 ∞ Then M is g r a d e d by D with respect to δ, and we call (D,δ)a grading of M. The image δ(x) is the degree of x. M is positively graded by D if in addition nonunits are mapped to nonunits; that is, δ(M • ) D D×. + ⊆ \ 5.2 Graded binoids 139

By deﬁnition, every grading δ : M • D maps units and idempotent elements of M to units and → • idempotent elements of D, respectively; that is, one always has δ(M ×) D× and δ((bool M) ) ⊆ ⊆ bool(D). Example 5.2.2. (1) Given a binoid M and a monoid D, the constant map x 0 is a D - grading of M, which is 7→ positive if and only if M is a binoid group. In particular, if M is commutative, δ(M) need not necessarily be contained in the center d D d + c = c + d for all c D of D. { ∈ | ∈ } (2) An integral binoid M is (D,δ) - graded if and only if δ : M • D is a monoid homomorphism. → (3) The blowup binoid associated to a commutative binoid M and an ideal M, cf. Remark 3.4.8, I ⊆ is an N - graded binoid via • δ : R N , (a; n) n . I −→ 7−→ This N - grading of RI is positive if and only if M is a binoid group. Remark 5.2.3. If δ : M • D is a D - grading of M, then M decomposes into the pointed union → M = M · d d[∈D of the pointed sets (M , ), where M = a M δ(a)= d , d D, with M + M M d ∞ d { ∈ | }∪{∞} ∈ d e ⊆ d+e for d, e D. By Proposition 3.4.7, ∈ K[M] = K[Md] dM∈D as K - modules, that is to say, K[M] is a D - graded K - algebra. In particular, if M is a positive binoid with a positive N - grading δ, then

a K[M] = Rn , where Rn := KT N nM∈ δ(Ma)=n is a positively graded K - algebra. Convention. In what follows, arbitrary binoids are assumed to be commutative. Lemma 5.2.4. Let M be a binoid, D a monoid, and δ : M • D a grading. → (1) Given an ideal , the quotient M/ is D - graded by [a] δ(a), a M . In particular, if M I I 7→ ∈ \ I is positively graded, then so is M/ . I (2) Every binoid homomorphism ϕ : N M with ker ϕ = yields a D - grading of N, namely → {∞} (δϕ)|N • . If δ is positive and ϕ local, then δϕ|N • is also a positive grading. (3) Every monoid homomorphism ϕ : D E yields an E - grading of M, namely ϕδ. If ϕ(D D×) → \ ⊆ E E×, then ϕδ is positive if δ is so. \ Proof. All statements are easily veriﬁed.

Lemma 5.2.5. Let n 1. A positively Nn - graded binoid contains only weakly separated elements. ≥ Proof. If there are any elements f,g M • with = f = f + g, then δ(f) = δ(f)+ δ(g), which ∈ ∞ 6 implies that δ(g)=0 (Nn)× since Nn is cancellative, hence g M × by the positivity of δ. ∈ ∈ Example 5.2.6. The binoid M := FC(x, y)/(2x = x + y = 3y) admits an unseparated element, namely f =2x +2y because f = f + y and y M × = 0 . Thus, M cannot be positively Nn - graded 6∈ { } for every n 1 by the preceding lemma. Indeed, the only Nn - grading δ is the trivial one with ≥ x, y 0 because 2δ(x) = δ(x)+ δ(y) implies that δ(x) = δ(y) by the cancellativity of Nn, and 7→ therefore 2δ(x)=3δ(y)=3δ(x). Hence, 0 = δ(x)= δ(y). 140 5 Separation and gradings

Theorem 5.2.7. Let M be a positive integral ﬁnitely generated binoid. The following conditions are equivalent. (1) M is separated. (2) M is positively Nk - graded for some k 1. ≥ (3) M is positively N - graded.

Proof. To prove (1) (2), we may assume that M is regular. To see this consider the finitely generated ⇒ regular binoid M = M/ • . By Lemma 5.1.14, M is again separated and positive, and can ∼can,M can by Lemma 5.2.4(2) every grading of Mcan extends to a grading of M by the canonical projection π : M M . Thus, let M be regular. By Proposition 1.14.6(2), there exists an embedding → can , ((ϕ : M ֒ (Nk Z/n Z Z/n Z)∞ , f ϕ(f) = (δ(f), ϕ (f),...,ϕ (f → × 1 ×···× ℓ 7−→ 1 ℓ where n ,...,n 2 and k 1. We claim δ : M • Nk is a positive grading of M. Since δ is a 1 ℓ ≥ ≥ → monoid homomorphism, we only need to show that δ(f)=0 Nk implies f = 0. Assume that ϕ(f)= ∈ (0, ϕ (f),...,ϕ (f)) for some f M. Then ϕ(mf) = (0, 0,..., 0) = ϕ(0) with m = n n , and 1 ℓ ∈ 1 ··· ℓ hence mf = 0 by the injectivity of ϕ, which implies f = 0 by the positivity of M. (2) implies (3) since the positive Nk - grading of M yields a positive N - grading of M via the evaluation homomorphism Nk N, (a ,...,a ) a + + a , cf. Lemma 5.2.4(3). (1) follows from (3) by Lemma 5.2.5. → 1 k 7→ 1 ··· k The binoid of Example 5.2.6 fulfills all conditions of the preceding theorem but is not separated and admits only the trivial Nn - grading, namely the constant map a 0. 7→ Example 5.2.8. The integrality condition in Theorem 5.2.7 cannot be ommited. To see this consider first the integral binoid M = FC(x,y,z)/( , , ), where R1 R2 R3 : a := y +2z = 2y + z =: b , R1 1 1 : a := x +2z = 2x + z =: b , R2 2 2 : a := x +3y = 2x + y =: b . R3 3 3 The binoid M fulfills all conditions of Theorem 5.2.7 but cannot be positively Nk -graded for k 1. In ≥ general, every grading δ : M D by a cancellative monoid D is trivial because and imply that → R1 R2 δ(x) = δ(y) = δ(z) and implies that 4δ(x)=3δ(x), hence 0 = δ(x) = δ(y) = δ(z). In particular, R3 M is not separated by Theorem 5.2.7. An unseparated element in M is given by

~~f := 4x +5y +6z = 2b1 +2a2 + b3~~

~~= 2a1 +2b2 + a3 = 5x +5y +6z = x + f .~~

Now the binoid M/(x + y + z = ) also fulﬁlls all conditions of Theorem 5.2.7 except the integrality, ∞ and for the same reason as M it cannot be Nk - graded, k 1, but it is separated. Elements of the ≥ form nx + mz and ny + mz, n,m 0, are separated because ≥ nx + mz = rx + sz or ny + mz = ry + sz if and only if r + s = n + m as one easily veriﬁes. The elements nx + my are separated since

~~x + ry = 2x + (r 2)y − = 3x + (r 4)y − sx +2y , if r =2s +2 ,s 1 , = 4x + (r 6)y = = ≥ − ··· (sx + y , if r =2s +1 ,s 1 . ≥ 5.2 Graded binoids 141~~

~~Though the absorbing element has no degree by deﬁnition, every grading δ : M • D of an arbitrary → binoid M gives rise to a well-deﬁned binoid homomorphism~~

~~x δ(x) , if x M •, θ : M M D∞ with θ(x)= ∧ ∈ −→ ∧ ( ∧ , otherwise, ∞ which in turn induces the following semigroup homomorphism~~

K-spec D∞ K-spec M K-spec M, (ϕ , ϕ ) (x ϕ (δ(x)) ϕ (x)) , × −→ 1 2 7−→ 7→ 1 · 2 cf. Proposition 1.5.6 and Proposition 1.8.12. In particular, if M is Nn -graded by δ : M • Nn, → a (δ (a),...,δ (a)), and K a field, there is a continuous An(K) - action on K-spec M =: X, 7→ 1 n namely An(K) X X, (t, ) t , × −→ Q 7−→ Q where t = (t ,...,t ) An(K) is given by ϕ : (Nn)∞ K with e t , and the K - point by 1 n ∈ (t) → i 7→ i Q ϕ : M K. Hence, t is the K - point Q → Q ϕ : a tδ(a)ϕ (a) = tδ1(a) tδn(a) ϕ (a) tQ 7−→ Q 1 ··· n · Q in X. Note that 1 = always holds. Q Q Definition 5.2.9. Let K be a field and M an Nn - graded binoid. With the above notation, a K - point is called An(K)- connected with another K - point if there is a t An(K) such that t = . Q P ∈ Q P Theorem 5.2.10. Let K a field. If M is a positive binoid that is positively Nn - graded, then every K - point is An(K) - connected with the special point. In particular, if M is also finitely generated, then every cancellative K - component of K- spec M contains the special point.

Proof. First, we claim that for an arbitrary K - point K-spec M the K - point 0 given by Q ∈ Q ϕ : a 0δ(aϕ (a) = 0δ1(a) 0δn(a) ϕ (a) , 0Q 7−→ Q ··· · Q is the special point; that is, ϕ0Q(a) = 1 if a = 0 and 1 otherwise. The ﬁrst is obvious because δ(0) = (0,..., 0) Nn and therefore ∈ ϕ (0) = 00 ϕ (0) = ϕ (0) = 1 . tQ · Q Q To show that ϕ vanishes on M 0 , note that δ(a) = (δ (a),...,δ (a)) = 0 Nn for every 0Q \{ } 1 n 6 ∈ a M 0 = M since M is positively graded. Thus for every i 1,...,r there is a j 1,...,n ∈ \{ } + ∈{ } ∈{ } such that δ (a) = 0 and therefore 0δj (a) = 0, which implies that ϕ (a) = 0. For the supplement, we j 6 0Q may assume that K is inﬁnite, otherwise consider the algebraic closure L of K and the diagram

An(K) K-spec M / K-spec M × ι ι An(L) L-spec M / L-spec M. × For an arbitrary K - point , the orbit O := t t An(K) of under the An(K)-operation on Q { Q | ∈ } Q X := K-spec M is given by the image of the continuous map An(K) X, t t . Hence, O is an n → 7→ Q irreducible subset of X because A (K) is so. By Proposition 4.4.7, X = i∈I Xi is the (ﬁnite) union of the closed cancellative K - components X , and the orbit O of has to be contained in one of these i Q S components since otherwise O = (O X ) would be a non-trivial decomposition of O. i∈I ∩ i S 142 5 Separation and gradings

~~Example 5.2.11. The group homomorphism id : Z/8Z Z/8Z deﬁnes a positive Z/8Z - grading of → the binoid (Z/8Z)∞. The C - spectrum is given by~~

~~C- spec(Z/8Z)∞.~~

~~Corollary 5.2.12. Let K be an inﬁnite ﬁeld and M = FC(x1,...,xn)/(f = g) so that~~

~~K[M]= K[X]/(Xf Xg) . − If supp f supp g = , then every cancellative component contains the special point. In particular, ∩ ∅ K- spec M is connected.~~

Proof. For f or/and g = , the statement is clear. So let f,g = . Then M satisﬁes all assumptions ∞ 6 ∞ of Theorem 5.2.7 and is separated since supp f supp g = . Hence, M is positively N - graded, and ∩ ∅ therefore every K - point is A1(K) - connected with the special point by Theorem 5.2.10. In particular, K-spec M is connected. 6 Binoids deﬁned by a simplicial complex and simplicial binoids

Simplicial complexes and their associated (Stanley-Reisner) algebras play a major role in combinatorial commutative algebra. Stanley-Reisner algebras are special monomial algebras and like every monomial algebra they can be realized as binoid algebras, cf. Remark 2.1.22 and Lemma 6.5.7. In this chapter, we treat these objects within the framework of binoid theory. We start with simplicial complexes by recalling basic definitions. With respect to the union and ∞ 0 intersection of faces, every simplicial complex ∆ defines two binoids, ∆∪ and ∆∩, which are usually very different as we show by studying their generating sets. The Zariski topology on their spectra can be described in detail and related to the simplicial complex ∆. For simplicial complexes ∆ and ∆˜ on V and V˜ , respectively, we study those morphisms V˜ V that induce canonical homomorphisms → between the binoids defined by ∆ and ∆˜ and between their N - spectra, which we introduce in the third section. Finally, we define the binoid M∆ associated to a simplicial complex ∆, whose binoid algebra is the Stanley-Reisner algebra of ∆, and characterize those binoids that yield Stanley-Reisner algebras.

~~Convention. In this chapter, arbitrary vertex sets are assumed to be ﬁnite.~~

~~6.1 Simplicial complexes~~

~~In this section, we generalize the binoid structures deﬁned on P(V ) to arbitrary simplicial complexes. For this we start with recalling basic deﬁnitions concerning them.~~

Deﬁnition 6.1.1. A(finite) simplicial complex ∆ on V is a subset-closed subset of P(V ) that contains all singletons v , v V . We refer to P(V ) as the f u l l simplicial complex on V . { } ∈ The elements of ∆ are called f a c e s and maximal faces under set inclusion are called f a c e t s . The dimension of a face F ∆ and that of the simplicial complex ∆ are deﬁned by ∈ dim F := #F 1 and dim ∆ := max dim F F ∆ , − { | ∈ } respectively. Elements of P(V ) ∆ are n o n fa c e s . A simplicial complex is p u r e if all its facets have \ the same dimension. Faces of dimension 0 and 1 are called v e r t i c e s and edges, respectively. We denote by fi(∆) the number of i - dimensional faces in ∆. Then the (d + 1) - tuple

~~f(∆) = (f−1(∆),f0(∆),...,fd(∆)) is called the f - vector of ∆, where d := dim ∆.~~

By deﬁnition, every simplicial complex on V is nonempty and contains the face , which has dimension ∅ 1. With respect to set inclusion, , v v V is the smallest simplicial complex on V and P(V ) is − {∅ { } | ∈ } the largest, and they coincide if and only if V is the empty set or a singleton. We have dim P( )= 1 ∅ − and dim P( v ) = 0. In general, 0 dim ∆ #V 1 for every simplicial complex ∆ on V with { } ≤ ≤ − #V 2, and dim ∆ = 0 if and only if ∆ = , v v V . Furthermore, f (∆) = #V and ≥ {∅ { } | ∈ } 0 f (∆) # facets of ∆ , where d = dim ∆, and equality holds if and only if ∆ is pure. d ≤ { } 144 6 Binoids deﬁned by a simplicial complex and simplicial binoids

~~A simplicial complex is uniquely determined by its facets since~~

~~∆ = P(F ) . F facet[ of ∆ The full simplicial complex P(V ) on V deﬁnes two binoids given by~~

P(V ) = (P(V ), ,V, ) and P(V ) = (P(V ), , , V ) , ∩ ∩ ∅ ∪ ∪ ∅ which have been studied in detail before. Their ideal lattices, ﬁltra, and spectra have been described in Example 2.1.10, Example 1.4.38, and Example 2.2.7, respectively. These results can be generalized to arbitrary simplicial complexes ∆ on V . Deﬁnition 6.1.2. For a simplicial complex ∆ on V denote by

~~∆ := (∆, , ) ∩ ∩ ∅ the boolean semibinoid with respect to the intersection of faces, and by~~

∆∞ := (∆ , , , ) ∪ ∪ {∞} ∪ ∅ ∞ the boolean binoid with respect to the union of faces, where F G := if this is not a face of ∆. ∪ ∞ The definition of ∆∩ is in line with our notation for the binoid P(V )∩ introduced in Example 1.1.3 because ∆∩ = P(V )∩ if ∆ = P(V ), while ∆∪ is only defined for ∆ = P(V ). However, we have the following identifications

0 ∆ = (∆ V , ,V, ) and ∆∞ = (∆ V , , , V ) , ∩ ∼ ∪{ } ∩ ∅ ∪ ∼ ∪{ } ∪ ∅ 0 0 which give rise to natural binoid embeddings ∆∩ ֒ P(V )∩ and ∆∪ ֒ P(V )∪ for every simplicial → 0 → ∞ complex ∆ P(V ). In particular, if ∆ = P(V ) V , then ∆∩ ∼= P(V )∩ and ∆∪ ∼= P(V )∪. Moreover, 0 ∞⊆ \{ } ∆∩ and ∆∪ are isomorphic to if V = , and to 0, if V = v . Let #V 2. Then ∆∩ ∞ {∞} ∅ { ∞} { } ≥ and ∆∪ are not cancellative since they contain non-trivial idempotent elements. The semibinoid ∆∩ is not integral and so is the binoid ∆∞ unless ∆ = P(V ). In general, ∆0 and ∆∞ are very different ∪ 6 ∩ ∪ binoids. To describe them in terms of generators, we need the following definition. Definition 6.1.3. Let M be a boolean commutative binoid. We say M is semifree up to idempotence if there exists a generating set a i I such that every element f M • can be { i | ∈ } ∈ written uniquely as f = a for a finite set J I. By abuse of notation, cf. Definition 1.3.12, i∈J i ⊆ (a ) is called a s e m i b a s i s of M and the set a i J the support of f, denoted by supp(f). i i∈I P { i | ∈ } ∞ Lemma 6.1.4. The binoid ∆∪ is semifree up to idempotence with semibasis v , v V . Conversely, { } ∈ ∞ if B is a finitely generated boolean binoid that is semifree up to idempotence, then B ∼= ∆∪ for some simplicial complex ∆. Proof. Only the converse is not trivial. Let V B be a finite generating set that is also a semibasis. ⊆ Obviously, ∆ := F V v = in B is a simplicial complex on V , and ∆∞ B with { ⊆ | v∈F 6 ∞ } ∪ → F v is a surjective binoid homomorphism, which is also injective since the elements of V 7→ v∈F P form a semibasis. Hence, B = ∆∞. P ∼ ∪ Example 6.1.5. 0 (1) Let ∆=P(V ) V . A semibasis of ∆∩ = P(V )∩ is given by the n facets V v , v V , and \{ }0 ∞ \{ } ∈ we have P(V )∪ = ∆∪ ∼= ∆∩ = P(V )∩ via P(V ) ∼ P(V ) , v V v . ∪ −→ ∩ { } 7−→ \{ } 6.1 Simplicial complexes 145

0 (2) Let ∆= , v v V . A semibasis of ∆∩ is given by the n facets v , v V , and we have 0 ∞{∅ { } | ∈ } { } ∈ ∆∪ = ∆∩ via ∼ 0 ∆∞ ∼ ∆ , v v . ∪ −→ ∩ { } 7−→ { } (3) Let V = 0,...,n 1 and let ∆ be the one-dimensional pure simplicial complex deﬁned by the { − } 0 n facets Fi := i,i +1 , i 0,...,n 2 , and Fn−1 := n 1, 0 . Then ∆∩ is semifree up to { } ∈{ − } { − } 0 ∞ idempotence with a semibasis given by these facets, and we have ∆∪ ∼= ∆∩ via ∼ 0 ∆∞ ∆ , i F . ∪ −→ ∩ { } 7−→ (k+i mod n) ∞ 0 In Theorem 6.1.14, we will show that ∆∪ and ∆∩ are isomorphic if and only if ∆ is composed of such simplicial complexes.

0 Proposition 6.1.6. Let ∆ be a simplicial complex on V . The binoid ∆∩ always admits a unique 0 minimal generating set X. Moreover, if ∆∩ is semifree up to idempotence, then X is given by the set of facets of ∆.

Proof. Obviously, the minimal generating set is given by the set consisting of all faces F ∆ for ∈ which there are no faces G, G′ ∆ F with F = G G′. For the supplement denote this set by ∈ \{ } ∩ 0 X. Since every semibasis is a minimal generating set, X has to be a semibasis if ∆∩ is semifree up to idempotence. However, X cannot be a semibasis if it contains a nonfacet. Indeed, if G X is a ∈ nonfacet and F ∆ a facet with G ( F , then G = F G are two diﬀerent expressions of G in terms ∈ ∩ of elements of X since the facets are always contained in X.

Example 6.1.7. The converse of the supplement of Proposition 6.1.6 need not be true. If ∆ is the simplicial complex on 1, 2, 3, 4 with facets 1, 2 , 1, 3 , 1, 4 , 2, 3 and 3, 4 , then the unique { 0 } { } { } { } { } { } minimal generating set of ∆∩ is given by the set of facets but it is not a semibasis since, for instance, 1, 2 1, 3 = 1, 2 1, 4 = 1, 3 1, 4 = 1 . { }∩{ } { }∩{ } { }∩{ } { } ∞ 0 Remark 6.1.8. In contrast to ∆∪ , the unique minimal generating set of ∆∩ need not be a semibasis. 0 If ∆∩ is not semifree up to idempotence, the minimal generating set has to be constructed. For the construction note that every subset X ∆0 that generates ∆0 has to contain the facets and the ⊆ ∩ ∩ faces F v , v F , F a facet, that cannot be generated by the facets (i.e. for which there is no facet \{ } ∈ G ∆ with G F = F v ). ∈ ∩ \{ } Now the unique minimal generating set is constructed in the following way: let dim F F ∆ { | ∈ facet = d ,...,d , where we may assume that dim ∆ = d > > d 0. We start with the set } { 1 s} 1 ··· s ≥ Y = F ∆ facet dim F = d . Since dim(F G) < min dim F, dim G for any two facets F, G ∆, 1 { ∈ | 1} ∩ { } ∈ the faces F v , v F , F Y , can only be generated by elements of Y , and those that cannot need \{ } ∈ ∈ 1 1 to be added to Y . Denote the resulting set by X and let Y := F ∆ facet dim F = d . The 1 1 2 { ∈ | 2} same argument as above shows that any face F v , v F , F Y , that is not contained in P(F ) \{ } ∈ ∈ 2 for F Y can only be generated by the facets F Y , otherwise they need to be added to X Y . ∈ 1 ∈ 2 1 ∪ 2 Denote the resulting set by X2 and proceed in the same manner till Xs is attained. By construction, Xs is the unique minimal generating set.

0 Since the unique minimal generating set of ∆∩ always contains the facets, it obviously may have strictly more than #V elements, but this may also happen if there are less than #V facets, which 0 ∞ 0 ∞ means that ∆∩ and ∆∪ cannot be isomorphic. Evidently, the binoids ∆∩ and ∆∪ may be very diﬀerent.

Example 6.1.9. Let ∆ = P(V ) be a simplicial complex on V deﬁned by pairwise disjoint facets 6 F1,...,Fr, r 2, and let n := #F1 + + #Fr = #V . In this case, the unique minimal generating 0 ≥ ··· set of ∆∩ has n+r elements, namely Fk and Fk v for v Fk, k 1,...,r , and is not a semibasis \{ 0} ∈ ∈{ } because F v = F F v . In particular, ∆ = ∆∞. k \{ } k ∩ k \{ } ∩ 6∼ ∪ 146 6 Binoids deﬁned by a simplicial complex and simplicial binoids

Lemma 6.1.10. Let I be ﬁnite, ∆ a simplicial complex on V , i I, and V := V the disjoint i i ∈ i∈I i union. Then the disjoint union U ∆ = (F ; k) F ∆ , k I P(V ) P(V ) i { k | k ∈ k \ {∅} ∈ }∪{∅} ⊆ i ∪{∅} ⊆ i]∈I i]∈I of (∆i)i∈I is a simplicial complex on V , where

~~0 ∞ 0 ∞ ∆i = : ∆i,∩ and ∆i = : ∆i,∪ . ∩ ∪ i]∈I i[∈I i]∈I i[∈I Also, the product~~

~~∆ = (F ) F ∆ ,i I P(V ) = P(V ) i { i i∈I | i ∈ i ∈ } ⊆ i ∼ Yi∈I Yi∈I of (∆i)i∈I is a simplicial complex on V , where~~

~~0 ∞ 0 ∞ ∆i = ∆i,∩ and ∆i = ∆i,∪ . ∩ ∪ Yi∈I Yi∈I Yi∈I i^∈I Proof. This is easily veriﬁed.~~

Recallthata partition ofaset V is a family (V ) of subsets such that V = V and V V = i i∈I i∈I i i ∩ j ∅ for i = j. Every simplicial complex ∆ on V admits a partition (V ) of V such that ∆ = ∆ , 6 i i∈I S i∈I i where ∆ is a simplicial complex on V , i I. Of course this partition need not be unique or might i i ∈ U be trivial; that is when #I = 1. However, there exists always a (unique) ﬁnest partition. Deﬁnition 6.1.11. A simplicial complex ∆ on V that admits no non-trivial partition of V is said to be connected. Proposition 6.1.12. Let ∆ be a simplicial complex on V . (1) ∆ is connected if and only if for any two vertices v, w V there are facets F ,...,F ∆ with ∈ 1 s ∈ v F and w F such that F F = for all i 1,...,s 1 . ∈ 1 ∈ s i ∩ i+1 6 ∅ ∈{ − } (2) There exists a unique partition (Vi)i∈I of V such that ∆ = i∈I ∆i, where ∆i is a connected simplicial complex on V , i I. i ∈ U Proof. Let A ∆ be the subset of facets of ∆ and consider the equivalence relation on V given by ⊆ v w : F ,...,F A such that v F , w F and F F = , i 1,...,s 1 . ∼ ⇔ ∃ 1 s ∈ ∈ 1 ∈ s i ∩ i+1 6 ∅ ∀ ∈{ − } Denote by V ,...,V , l 1, the equivalence classes. Now (1) translates to: ∆ is connected if and only 1 l ≥ if l = 1. Clearly, if ∆ is not connected, then l 2. Conversely, if l 2 then (V ) , I = 1,...,l , is a ≥ ≥ i i∈I { } non-trivial partition of V such that ∆ = ∆ , where ∆ is the simplicial complex F ∆ F V i∈I i i { ∈ | ⊆ i} on V , i I. This also proves (2) since the ∆ are connected by (1). i ∈ U i Example 6.1.13. The full simplicial complex P(V ) and the simplicial complex of Example 6.1.5 (3) are connected, whereas ∆ = , v v V is not connected for #V 2 since ∆ = P( v ). {∅ { } | ∈ } ≥ v∈V { } 0 Theorem 6.1.14. Let ∆ be a simplicial complex on V . The binoid ∆∩ is semifree upU to idempotence if and only if ∆ decomposes into connected simplicial complexes of the form

∆(1) := P(W ) W or ∆(2) := P( v ) or ∆(3) := , i , i,i + 1 mod n i 0,...,n 1 . \{ } { } {∅ { } { } | ∈{ − }} ∞ 0 In this case (and only in this case), ∆∪ and ∆∩ are isomorphic. 6.2 Simplicial morphisms 147

0 Proof. Of course, ∆∩ is semifree up to idempotence if it decomposes in such simplicial complexes because they are semifree up to idempotence by Example 6.1.5 and then the union of their semibases 0 yields a semibasis of ∆∩. Moreover, if V = i Vi and ∆ = i∈I ∆i, where ∆i is one of these three simplicial complexes on V , i I, the isomorphism i ∈ U U 0 0 ∆∞ = : ∆∞ ∼ : ∆ = ∆ ∪ i,∪ −→ i,∩ ∩ i[∈I i[∈I ∞ 0 is given componentwise ∆i,∪ ∆i,∩ as in Example 6.1.5. For the converse, it suﬃces to show that 0 → (r) if ∆∩ is connected and semifree up to idempotence, then ∆ is of the form ∆ for r = 1, 2, or 3. If #V 2, there is nothing to show. So let #V 3. Note that every zero-dimensional simplicial complex ≤ ≥ is of the form ∆(2). Furthermore, every one-dimensional connected simplicial complex that is semifree up to idempotence is of the form ∆(3). Indeed, in such a simplicial complex every facet need to have dimension 1 by the connectedness, and every vertex has to be contained in precisely two facets because the simplicial complex is semifree up to idempotence. Thus, we may assume that dim ∆ =: d 2. We ≥ need to show that ∆ = P(V ) V . Denote by A the set of facets of ∆. By Proposition 6.1.6, A is a 0 \{ } semibasis of ∆ . In particular, given a facet F of dimension d, say F = v i I , I = 1,...,d+1 , ∩ { i | ∈ } { } there are facets F ,...,F A with (after a suitable renumbering) F F = F v and dim F = d, 1 d+1 ∈ ∩ i \{ i} i i I. Hence, F = F v w for some w V F , i I. For for i = j this gives ∈ i \{ i}∪{ i} i ∈ \ ∈ 6

F vi, vj , if wi = wj , Fi Fj = \{ } 6 ∩ ((F vi, vj w , if wi = wj =: w . \{ }⊎{ } In the ﬁrst case, one has F F F = F F , which is a contradiction to A being a semibasis. Hence, ∩ i ∩ j i ∩ j w = w for all i I and therefore P(W ) W ∆ with W := v ,...,v , w V . To show that i ∈ \{ }⊆ { 1 d+1 }⊆ W = V suppose that v V W . Since ∆ is connected we ﬁnd by Proposition 6.1.12 for any w W ∈ \ ∈ facets G ,...,G A with w G , v G , and G G = , j 1,...,s 1 . We may assume 1 s ∈ ∈ 1 ∈ s j ∩ j+1 6 ∅ ∈{ − } that G W ; otherwise extend the sequence G ,...,G by a facet G W with w G . Then there 1 ⊆ 1 s 0 ⊆ ∈ 0 is an r 1,...,s 1 such that G W but G W . This is again a contradiction to A being a ∈{ − } r ⊆ r+1 6⊆ semibasis because = G G W is also the intersection of certain F from above. Thus, V = W ∅ 6 r ∩ r+1 ⊆ i and therefore ∆ = P(V ) V . \{ }

~~6.2 Simplicial morphisms~~

In this section, we consider morphisms V˜ V between vertex sets of simplicial complexes ∆˜ and ∆ → and investigate when such morphisms induce homomorphisms between the binoids deﬁned by these simplicial complexes. This yields the deﬁnition of simplicial, α-simplicial, and β-simplicial morphisms for which we will give various examples that will appear later on.

Lemma 6.2.1. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively, and λ : V˜ V a → map. The following statements are equivalent. (1) Every image of a nonface is a nonface; that is, F ∆˜ implies λ(F ) ∆. 6∈ 6∈ (2) Every preimage of a face is a face; that is, F ∆ implies λ−1(F ) ∆˜ . ∈ ∈ Proof. (1) (2) Let F ∆. Since λ(λ−1(F )) F the image λ(λ−1(F )) is also a face of ∆. ⇒ ∈ ⊆ Hence, λ−1(F ) ∆˜ since otherwise λ(λ−1(F )) ∆ by assumption on λ. (2) (1) Let F ∆.˜ ∈ 6∈ ⇒ 6∈ Since F λ−1(λ(F )) the preimage λ−1(λ(F )) is not a face of ∆.˜ Hence, λ(F ) ∆ since otherwise ⊆ 6∈ λ−1(λ(F )) ∆˜ by assumption on λ. ∈ 148 6 Binoids deﬁned by a simplicial complex and simplicial binoids

Deﬁnition 6.2.2. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively, and λ : V˜ V a → map. (1) λ is a s i m p l i c i a l morphism if every image of a face is a face; that is, F ∆˜ implies λ(F ) ∆. ∈ ∈ (2) λ is an α -simplicial morphism if λ satisﬁes one of the equivalent conditions of Lemma 6.2.1. (3) λ is a β -simplicial morphism if every preimage of a nonface is a nonface; that is, F ∆ 6∈ implies λ−1(F ) ∆.˜ 6∈ Since these properties depend on the chosen simplicial complexes, we will write

~~λ : (V,˜ ∆)˜ (V, ∆) . −→ The category of simplicial complexes together with simplicial, α-simplicial and β-simplicial morphisms will be denoted by SC, SCα, and SCβ, respectively.~~

~~Our deﬁnition of a simplicial morphism is consistent with the terminology in topology of abstract simplicial complexes, cf. [25, Chapter II.2].~~

Example 6.2.3. Here are examples of morphisms that satisfy only one property, simplicial or α- simplicial or β-simplicial, but none of the other both. For this let V = 1,...,n , n 3. { } ≥ (1) Let T = V n and F = n 1,n . The map \{ } { − } λ : (T, P(T )) (V, P(V ) V, F ) with v v −→ \{ } 7−→ is simplicial because P(T ) P(V ) V, F . Since λ−1(F ) = n 1 P(T ) T but F ⊆ \{ } { − } ∈ \{ } 6∈ P(V ) V, F it is not α-simplicial, and it is not β-simplicial because λ−1(V )= T P(T ) but \{ } ∈ V P(V ) V, F . 6∈ \{ } (2) Let ∆= , v v V n . The map {∅ { } | ∈ \{ }} v , if v = n , λ : (V, P(V ) V ) (V n , ∆) with v 6 \{ } −→ \{ } 7−→ (n 1 , otherwise, − is α-simplicial because V is the only nonface of P(V ) V and λ(V ) = V ∆. On the other \{ } 6∈ hand, it is not simplicial because λ( 1, 2 ) = 1, 2 ∆ but 1, 2 P(V ) V , and since { } { } 6∈ { } ∈ \{ } λ−1( 1, 2 )= 1, 2 P(V ) V but 1, 2 ∆ it is not β-simplicial. { } { } ∈ \{ } { } 6∈ (3) Let n =3, ∆=P(V ) V, 1, 3 , and ∆=˜ , 1 , 2 . The map \{ { }} {∅ { } { }} 1, if v 1, 3 , λ : (V, ∆) ( 1, 2 , ∆)˜ with v ∈{ } −→ { } 7−→ (2, otherwise,

is β-simplicial because the only nonface in ∆˜ is 1, 2 and λ−1( 1, 2 )= V ∆. On the other { } { } 6∈ hand, it is not simplicial because λ( 1, 2 )= 1, 2 ∆˜ but 1, 2 ∆, and it is not α-simplicial { } { } 6∈ { } ∈ because λ−1( 1 )= 1, 3 ∆ but 1 ∆.˜ { } { } 6∈ { } ∈ Example 6.2.4. Here are examples of morphisms that satisfy all except one property. Let V = 1,...,n with n 2. { } ≥ (1) Let ∆ and ∆˜ be a simplicial complexes on V such that ∆˜ ( ∆. Then

id : (V, ∆)˜ (V, ∆) −→ is simplicial since ∆˜ ∆. It is β-simplicial since nonfaces of ∆ are nonfaces of ∆,˜ but it is not ⊆ α-simplicial because there is by assumption a face F ∆ with F ∆,˜ hence λ−1(F )= F ∆.˜ ∈ 6∈ 6∈ 6.2 Simplicial morphisms 149

~~(2) Let T ( V . The morphism~~

λ : (T, P(T )) (V, P(V ) V ) with v v −→ \{ } 7−→ is simplicial since P(T ) P(V ), and it is α-simplicial since P(T ) has no nonfaces. However, λ ⊆ is not β-simplicial because λ−1(V )= T P(T ) but V P(V ) V . ∈ 6∈ \{ } (3) Let T = V n . The morphism \{ } v , if v = n , λ : (V, P(V ) V ) (T, P(T ) T ) with v 6 \{ } −→ \{ } 7−→ (n 1 , otherwise, − is α-simplicial since V is the only nonface of P(V ) V and λ(V )= T P(T ) T , and since T \{ } 6∈ \{ } is the only nonface of P(T ) T and λ−1(T )= V P(V ) V it is also β-simplicial. However, \{ } 6∈ \{ } λ is not simplicial because λ(T )= T P(T ) T but T P(V ) V . 6∈ \{ } ∈ \{ } Lemma 6.2.5. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively. (1) If λ : (V,˜ ∆)˜ (V, ∆) is surjective and simplicial, then it is β-simplicial. → (2) If λ : (V,˜ ∆)˜ (V, ∆) is injective and β-simplicial, then it is simplicial. → Proof. (1) If λ is surjective, we have F = λ(λ−1(F )) for all F V . Thus, if λ is simplicial and ⊆ λ−1(F ) ∆,˜ then λ(λ−1(F )) ∆ and therefore F ∆. (2) If λ is injective we have F = λ−1(λ(F )) ∈ ∈ ∈ for all F V˜ . Thus, if λ is β-simplicial and λ(F ) ∆, then F = λ−1(λ(F )) ∆.˜ ⊆ 6∈ 6∈ Example 6.2.6. Let V = 1,...,n 1 , n 2. Consider ∆ = P(F ) P(F ), where F = 1,...,k { − } ≥ 0 ∪ 1 0 { } and F = s,...,n 1 for some s 2,...,k +1 , and ∆=P(˜ F ) P(F n ). The surjection 1 { − } ∈{ } 0 ∪ 1 ∪{ } v , if v = n , λ : (V n , ∆)˜ (V, ∆ ) with v 6 ∪{ } −→ } 7−→ (n 1 , otherwise, − is simplicial and β-simplicial but not injective. On the other hand, for ∆ = , v v V and 0 {∅ { } | ∈ } ∆˜ = , v , v,n v V n the injective map 0 {∅ { } { } | ∈ ∪{ }} λ : (V, ∆ ) (V n , ∆˜ ) with v v 0 −→ ∪{ } 0 7−→ is simplicial and β-simplicial but not surjective. Note that both morphisms are α-simplicial. Lemma 6.2.7. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively. (1) If λ : (V,˜ ∆)˜ (V, ∆) is simplicial, then −→ , if F = , ∅ ∅ ∞ ˜ ∞ −1 ϕλ : ∆∪ ∆∪ with F  , if F = or λ (v)= for some v F, −→ 7−→ ∞ ∞ ∅ ∈ λ−1(F ) , otherwise,  is a binoid homomorphism, which is surjective if λ is injective. (2) λ : (V,˜ ∆)˜ (V, ∆) is α-simplicial if and only if −→ , if F = or λ(F ) ∆ , α ˜ ∞ ∞ ϕλ : ∆∪ ∆∪ with F ∞ ∞ 6∈ −→ 7−→ (λ(F ) , otherwise,

is a binoid homomorphism, which is surjective if λ is so. If λ is α-simplicial, simplicial, and α injective, then ϕλ is also injective. 150 6 Binoids deﬁned by a simplicial complex and simplicial binoids

~~(3) λ : (V,˜ ∆)˜ (V, ∆) is β-simplicial if and only if −→ −1 ˜ β ∞ ˜ ∞ , if F = or λ (F ) ∆ , ϕλ : ∆∪ ∆∪ with F ∞ ∞ 6∈ −→ 7−→ (λ−1(F ) , otherwise,~~

~~is a binoid homomorphism, which is surjective if λ is bijective.~~

Proof. (1) To verify that ϕ is well-defined, we need to check that ϕ (F G) = (or equivalently λ λ ∪ ∞ λ−1(F G) ∆),˜ whenever F G ∆ for F, G ∆ and λ−1(v) = for all v F G. Indeed, the ∪ 6∈ ∪ 6∈ ∈ 6 ∅ ∈ ∪ latter assumption yields λ(λ−1(F G)) = F G. Hence, λ−1(F G) ∆˜ since otherwise, we had ∪ ∪ ∪ 6∈ F G ∆ because λ is simplicial. Moreover, ϕ is a binoid homomorphism because λ−1(F G) = ∪ ∈ λ ∪ λ−1(F ) λ−1(G) for all F, G ∆, and ϕ ( ) = and ϕ ( ) = by definition. If λ is injective, ∪ ∈ λ ∞ ∞ λ ∅ ∅ then G = λ−1(λ(G)) for all G V˜ . In particular, if G ∆,˜ then λ(G) ∆ since λ is simplicial, and −1 ⊆ ∈ ∈ hence ϕλ(λ(G)) = λ (λ(G)) = G, which shows that ϕλ is surjective if λ is injective. α (2) Clearly, ϕλ is a well-defined binoid homomorphism if λ is α-simplicial. For the converse, we show that every image of a nonface is a nonface. So let F ∆.˜ This means F = in ∆˜ ∞, and hence 6∈ ∞ ∪ λ(F ) = λ(v) = ϕα( v ) = ϕα(F ) = ϕα( )= { } { λ { } } λ λ ∞ ∞ v[∈F v[∈F ∞ α ∞ in ∆∪ because ϕλ is a binoid homomorphism. Thus, λ(F ) ∆. If λ is surjective and F ∆∪ , then α −1 −1 α 6∈ ∈ ϕλ (λ (F )) = λ(λ (F )) = F , which shows that ϕλ is also surjective. For the supplement note that α ∞ ˜ ∞ since λ is simplicial and α-simplicial, we have ϕλ (F )= in ∆∪ if and only if F = in ∆∪ . So we ˜ α α ∞ ∞ only need to consider F, G ∆ with ϕλ (F ) = ϕλ (G) for the injectivity. Then λ(F ) = λ(G), which ∈ α implies F = G by the injectivity of λ. Thus, ϕλ (F ) = is also injective. β β (3) Clearly, ϕλ is a well-defined binoid homomorphism if λ is β-simplicial. Conversely, if ϕλ is a binoid homomorphism and F ∆, then F = in ∆∞, and hence ϕβ(F )= ϕβ( )= . This shows 6∈ ∞ ∪ λ λ ∞ ∞ that λ−1(F ) ∆,˜ since otherwise ϕβ(F )= λ−1(F ) = in ∆˜ ∞. If λ is bijective, then G = λ−1(λ(G)) 6∈ λ 6 ∞ ∪ for every G ∆.˜ This proves the surjectivity of ϕβ because λ(G) ∆ by Lemma 6.2.5(2). ∈ λ ∈

~~Remark 6.2.8.~~

(1) For a simplicial morphism λ that is surjective but not injective ϕλ need not be surjective as ∞ ˜ ∞ Example 6.2.6 shows. Here F1 has no preimage under ϕλ : ∆∪ ∆∪ because every G ∆ −1 −1 → ∈ with F1 = ϕλ(G) = λ (G) fulﬁlls n 1 G, but then n λ (G), which is not contained in −1 − ∈ ∈ F1. Hence, F1 ( λ (G). (2) Example 6.2.4(2) shows that ϕβ is usually no binoid homomorphism, whether λ : V˜ V is λ → simplicial or α-simplicial. β (3) If λ is β-simplicial, then ϕλ need not be surjective if λ is surjective but not injective. An example is given by the β-simplicial morphism λ : (V, ∆) ( 1, 2 , ∆)˜ of Example 6.2.3(3). Here the → { } facet 1, 2 ∆ has no preimage under ϕβ : ∆˜ ∞ ∆∞. { } ∈ λ ∪ → ∪ Lemma 6.2.9. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively, and λ : V˜ V a → map. (1) If λ : (V,˜ ∆)˜ (V, ∆) is simplicial and injective, then →

~~˜ 0 0 0 , if F =0 , ψλ : ∆∩ ∆∩ with F −→ 7−→ (λ(F ) , otherwise,~~

~~is a binoid embedding. 6.2 Simplicial morphisms 151~~

~~(2) If λ : (V,˜ ∆)˜ (V, ∆) is α-simplicial, then →~~

~~α 0 ˜ 0 0 , if F =0 , ψλ : ∆∩ ∆∩ with F −→ 7−→ (λ−1(F ) , otherwise,~~

~~is a binoid homomorphism.~~

Proof. For (1) note that λ(F G) = λ(F ) λ(G) for all F, G V˜ if and only if λ is injective. ∩ ∩ ⊆ Therefore, the injectivity is necessary. The rest of the lemma follows from the deﬁnitions of simplicial and α-simplicial morphisms.

Remark 6.2.10. For λ being only injective, the map ψ : ∆˜ 0 ∆0 with ∩ → ∩ 0 , if F =0 or λ(F ) ∆ , F 6∈ 7−→ (λ(F ) , otherwise, is no binoid homomorphism if there is an F ∆˜ with F = V˜ and λ(F ) ∆ because for v V˜ F ∈ 6 6∈ ∈ \ one has λ(v) ∆, and hence ∈ ψ(F v ) = ψ( ) = but ψ(F ) ψ( v ) = ψ( v ) = λ(v) . ∩{ } ∅ ∅ ∩ { } { } Similarly, the map ψ : ∆0 ∆˜ 0 with ∩ → ∩ 0 , if F =0 or λ−1(F ) ∆˜ , F 6∈ 7−→ (λ−1(F ) , otherwise, is no binoid homomorphism if λ is not α-simplicial. Take, for instance, ∆ = P(V ), ∆=˜ , v v {∅ { } | ∈ V , and λ = id . For pairwise diﬀerent elements u,v,w V one has } V ∈ ψ( u, w v, w ) = ψ( w ) = w but ψ( u, w ) ψ( v, w )=0 0=0 . { }∩{ } { } { } { } ∩ { } ∩ We close this section by applying Lemma 6.2.7 and Lemma 6.2.9 to three important examples that will appear later on.

Example 6.2.11. Let ∆ and ∆˜ be simplicial complexes on V such that ∆˜ ∆. The morphism ⊆ id : (V, ∆) (V, ∆)˜ −→ is α-simplicial (and not simplicial or β-simplicial if ∆˜ ( ∆). We thus get the binoid epimorphism

˜ α ∞ ˜ ∞ , if F ∆ , ϕid : ∆∪ ∆∪ with F ∞ 6∈ −→ 7−→ (F , otherwise, and the binoid embedding 0 0 ψα : ∆˜ ∆ with v v . id ∩ −→ ∩ 7−→ Both binoid homomorphisms can also be deduced form the identity

id : (V, ∆)˜ (V, ∆) , −→ which is simplicial and β-simplicial (bute not α-simplicial if ∆˜ ( ∆), cf. Example 6.2.4(1). Note that β α here ϕ and ϕid coincide with ϕid. ide e 152 6 Binoids deﬁned by a simplicial complex and simplicial binoids

Example 6.2.12. Let ∆ be a simplicial complex on V . If T V and ∆(T ) = F ∆ F T , ⊆ { ∈ | ⊆ } then ι : (T, ∆(T )) (V, ∆) with v v −→ 7−→ is simplicial and α-simplicial (and not β-simplicial if T ( V , see below). We thus get the binoid epimorphism

~~∞ ∞ F , if F ∆(T ) , ϕι : ∆∪ ∆(T )∪ with F ∈ −→ 7−→ ( , otherwise, ∞ and the binoid embedding~~

ϕα : ∆(T )∞ ∆∞ with F F. ι ∪ −→ ∪ 7−→ In particular, we have ϕα ϕ ∆(T )∞ ι ∆∞ ι ∆(T )∞ ∪ −→ ∪ −→ ∪ α with ϕιϕι = id. Moreover, we have the binoid embedding

~~0 0 ψ : ∆(T ) ∆ with F F ι ∩ −→ ∩ 7−→ and the binoid epimorphism~~

α 0 0 0 , if F =0 , ψι : ∆∩ ∆(T )∩ with F −→ 7−→ (F T , otherwise. ∩ If T ( V , then ι is not β-simplicial unless ∆ = P(V ). To see this take a facet F ∆(T ). Since 6 ∈ ∆ = P(V ) there are v ,...,v V T such that G := F v ,...,v ∆ but ι−1(G)= F ∆(T ). 6 1 r ∈ \ ∪{ 1 r} 6∈ ∈ If ∆=P(V ), the map ι is also β-simplicial, hence

β ∞ ∞ , if F = , ϕι : P(V )∪ ∆(T )∪ with F ∞ ∞ −→ 7−→ (F T , otherwise, ∩ is a binoid homomorphism which is diﬀerent from ϕ : P(V )∞ ∆(T )∞ because ϕ ( v ) = v = ι ∪ → ∪ ι { } { } ϕβ( v ) for v T , but if v T , then ι { } ∈ 6∈ ϕ ( v )= and ϕβ ( v )= . ι { } ∞ ι { } ∅

Example 6.2.13. Let ∆ be a simplicial complex on V , i I, where I is ﬁnite and V V = for i i ∈ i ∩ j ∅ i = j. For every k I, the injective morphism 6 ∈ ι : V , ∆ V , ∆ with v (v; k) k k k −→ i i 7−→ i]∈I i]∈I is simplicial and α-simplicial. We thus get the binoid epimorphism

~~, if i = k , : ∞ ∞ ϕιk : ∆i,∪ ∆k,∪ with (F ; i) ∞ 6 −→ 7−→ (F , otherwise, i[∈I and the binoid embedding~~

~~ϕα : ∆∞ : ∆∞ with F (F ; k) . ιk k,∪ −→ i,∪ 7−→ i[∈I 6.3 N - points of a simplicial complex 153~~

~~Moreover, we have the binoid embedding~~

~~0 0 ψ : ∆ : ∆ with F (F ; k) ιk k,∩ −→ i,∩ 7−→ i[∈I and the binoid epimorphism~~

~~0 , if (F,i)=0 , α : 0 0 ψιk : ∆i,∩ ∆k,∩ with (F ; i) (F ; k) , if i = k , −→ 7−→  i[∈I  , otherwise. ∅  The injective morphism ~~

˜ι : V , ∆ V , ∆ with v (v; k) k k k −→ i i 7−→ i]∈I Yi∈I is only α-simplicial for #I 2. Here we obtain the binoid embedding ≥ ϕα : ∆∞ ∆∞ with F F , ˜ιk k,∪ −→ i,∪ 7−→∅∧···∧∅∧ ∧∅∧···∧∅ i^∈I where F is the kth entry, and the binoid epimorphism

~~α 0 0 ψ : ∆ ∆ with i∈I Fi Fk . ˜ιk i,∩ −→ k,∩ ∧ 7−→ i^∈I~~

~~6.3 N -points of a simplicial complex~~

This section deals with N - points of a simplicial complex ∆, which will later be related to the N - points of the (simplicial) binoid deﬁned by ∆ Convention. In this section, N always denotes a commutative nonzero binoid. Deﬁnition 6.3.1. Let ∆ be a simplicial complex on V . An N - point of∆isamap

ρ : V N such that ρ(v)= if A ∆ . −→ ∞ 6∈ vX∈A The N - s p e c t r u m of ∆ is the set of all N - points of ∆, denoted by N-spec∆. With respect to the addition N- spec ∆ is a semibinoid with absorbing element ε : v , v V . 7→ ∞ ∈ If N is integral the property ρ(v) = for all A ∆ is equivalent to: for every A ∆ there v∈A ∞ 6∈ 6∈ is a v A such that ρ(v) = , and if N is a multiplicatively written, this condition on ρ : V N ∈ ∞P → translates to ρ(v) = 0 for every nonface A of ∆. In particular, if K is a field, the map ρ : V K v∈A → is a K - point if and only if for every A ∆ there is a v A such that ρ(v) = 0. Later we will prove Q 6∈ ∈ that K-spec∆ ∼= K-Spec K[∆], where K[∆] is the Stanley-Reisner algebra of ∆, cf. Proposition 6.5.16. Lemma 6.3.2. Let ∆ be a simplicial complex on V . A subset F V is a face of ∆ if and only if ⊆ .(χ : V 0, is an N - point (with respect to 0, ֒ N F →{ ∞} { ∞} → Proof. Obviously, F is a face if χ is an N -point because χ (v)=0 = . Conversely, if F v∈F F 6 ∞ F ∆ and A ∆, then A F . Thus, we find a w A (A F ), but this means χ (w) = and ∈ 6∈ 6⊆ ∈ \ P∩ F ∞ therefore χ (v)= . v∈A F ∞ P 154 6 Binoids defined by a simplicial complex and simplicial binoids

Proposition 6.3.3. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively. The following conditions on a map λ : V˜ V are equivalent. → (1) λ : (V,˜ ∆)˜ (V, ∆) is an α-simplicial morphism. → (2) For every ρ N- spec∆, N a commutative binoid, ρλ : V˜ N is an N - point of ∆˜ . ∈ → (3) For every ρ K- spec∆, K a ﬁeld, ρλ : V˜ K is a K - point of ∆˜ . ∈ → In particular, every α-simplicial morphism λ : (V,˜ ∆)˜ (V, ∆) induces a semibinoid homomorphism → λ∗ : N- spec∆ N- spec ∆˜ −→ with ρ ρλ, which is injective if λ is surjective. 7→ Proof. (1) (2) For an arbitrary nonface A V˜ of ∆,˜ we have ⇒ ⊆

ρλ(v) = nwρ(w) vX∈A w∈Xλ(A) for certain n 1, w λ(A). By assumption, λ(A) ∆ and ρ is an N - point, hence ρ(w)= w ≥ ∈ 6∈ w∈λ(A) , but this is a summand of ρλ(v). (2) (3) is obvious. (3) (1) Suppose that A V˜ is a ∞ v∈A ⇒ ⇒ P ⊆ nonface of ∆˜ but λ(A) ∆. By Lemma 6.3.2, the map χ : V 1, 0 K is a K - point of ∆, ∈ P λ(A) →{ }⊆ which yields the K - point χ λ of ∆˜ with χ (λ(v)) = 1, a contradiction to A ∆.˜ The λ(A) v∈A λ(A) 6∈ supplement is clear. Q Example 6.3.4. The α-simplicial morphism id : (V, ∆) (V, ∆)˜ from Example 6.2.11, where ∆ and → ∆˜ are simplicial complexes on V such that ∆˜ ∆, induces a semibinoid embedding ⊆ N-spec ∆˜ N- spec ∆ with ρ ρ . −→ 7−→ The α-simplicial morphisms ι : (T, ∆(T )) (V, ∆) from Example 6.2.12, where T V and ∆(T )= → ⊆ F ∆ F T , induces the semibinoid homomorphism { ∈ | ⊆ } N-spec∆ N-spec∆(T ) with ρ ρ . −→ 7−→ |T Finally, the α-simplicial morphisms ι : (V , ∆ ) V, ∆ and ˜ι : (V , ∆ ) V, ∆ , k k k → i∈I i k k k → i∈I i k I, from Example 6.2.13, where ∆ is a simplicial complex on V , i I, and V = V , induce ∈ i U i ∈ i∈I Qi the semibinoid homomorphisms U N-spec ∆ N-spec∆ with ρ ρ i −→ k 7−→ |(Vk;k) i]∈I and N-spec ∆ N-spec∆ with ρ ρ . i −→ k 7−→ |(Vk ;k) Yi∈I Proposition 6.3.5. Let ∆ and ∆˜ be simplicial complexes on V and V˜ , respectively. The following conditions on a map λ : V˜ V are equivalent. → (1) λ : (V,˜ ∆)˜ (V, ∆) is a β-simplicial morphism. → (2) For every ρ N- spec ∆˜ , N a commutative binoid, the map ∈ ρ : V N with v p(w) −→ 7−→ −1 w∈Xλ (v) b is an N - point of ∆. 6.3 N - points of a simplicial complex 155

(3) For every ρ K- spec ∆˜ , K a ﬁeld, the map ∈ ρ : V K with v p(w) −→ 7−→ −1 w∈λY (v) b is a K - point of ∆. In particular, every β-simplicial morphism λ : (V,˜ ∆)˜ (V, ∆) induces a semibinoid homomorphism → λ∗ : N- spec ∆˜ N- spec∆ −→ with ρ ρ, which is injective if λ is so. 7→ Proof. (1)b (2) For an arbitrary nonface A V of ∆, we have ⇒ ⊆ ρ(v) = ρ(w) = ρ(w) = −1 −1 ∞ vX∈A vX∈A w∈Xλ (v) w∈λX(A) b because ρ is an N - point of ∆˜ and λ−1(A) ∆˜ by assumption on λ. Hence, ρ is an N - point of ∆. 6∈ (2) (3) is obvious. To show (3) (1) suppose that A V is a nonface of ∆ but λ−1(A) ∆.˜ By ⇒ ⇒ ⊆ ∈ Lemma 6.3.2, the map χ −1 : V˜ 1, 0 K is a K - point of ∆,˜ so by assumptionb λ (A) →{ }⊆

~~χλ(A) : V K with v χλ−1(A)(w) , −→ 7−→ −1 w∈λY (v) b is a K - point of ∆ with~~

χλ(A)(v) = χλ−1(A)(w) = χλ−1(A)(w)=1 , −1 −1 vY∈A vY∈A w∈λY (v) w∈λY(A) b which is a contradiction to A ∆. For the supplement assume that ρ ,ρ N-spec ∆˜ such that 6∈ 1 2 ∈ ρ = ρ . Then ρ (v)= ρ (v) for all v V , but by the injectivity of λ we have 1 2 1 2 ∈ −1 b b b b ρi(v) = ρi(w) = ρi(λ (v)) −1 w∈Xλ (v) b for all v V , i 1, 2 . Thus, ρ = ρ . ∈ ∈{ } 1 2

Example 6.3.6. The β-simplicial morphism id : (V, ∆)˜ (V, ∆) from Example 6.2.11, where ∆ −→ and ∆˜ are simplicial complexes on V such that ∆˜ ∆, induces a semibinoid embedding e ⊆ N-spec ∆˜ N- spec ∆ with ρ ρ , −→ 7−→ which is the semibinoid embedding of Example 6.3.4. In particular, we always have an embedding

N-spec ∆˜ N-specP(V )= N #V . −→ Remark 6.3.7. By Lemma 6.2.5(1), a surjective simplicial morphism λ : (V,˜ ∆)˜ (V, ∆) is also → β-simplicial. Hence, Proposition 6.3.5 applies to λ in this case and gives a semibinoid embedding

~~λ∗ : N-spec ∆˜ N- spec ∆ with ρ ρ . −→ 7−→ b 156 6 Binoids deﬁned by a simplicial complex and simplicial binoids~~

~~6.4 Topology of simplicial complexes~~

∞ 0 In this section, we study the ideal lattices of ∆∪ and ∆∩ and describe the spectra of these binoids. If ∆ is not the full simplicial complex on V (i.e. ∆ = P(V )), we will use the identiﬁcation 6 0 ∆ = (∆ V , ,V, ) and ∆∞ = (∆ V , , , V ) ∩ ∼ ∪{ } ∩ ∅ ∪ ∼ ∪{ } ∪ ∅ throughout this section. For a simplicial complex ∆ = P(V ) on V , the canonical binoid embedding 6 and epimorphism induced by the α-simplicial morphism id : (V, P(V ) V ) (V, ∆), cf. Example \{ } → 6.2.11, read as follows

~~0 ∞ ι∆ : ∆ P(V ) and π∆ : P(V ) ∆ ∩ −→ ∩ ∪ −→ ∪ F , if F ∆, F F F ∈ 7−→ 7−→ (V , otherwise.~~

Proposition 6.4.1. Let ∆ = P(V ) be a simplicial complex on V . 6 (1) ∆ is an ideal in P(V )∩. (2) A subset A P(V ) is a prime ideal in P(V ) if and only if A = P(V ) deﬁnes a pure simplicial ⊆ ∩ 6 complex on V of dimension #V 2; that is, A = P(V v ) for some = G V . − v∈G \{ } ∅ 6 ⊆ c c ∞ ∞ (3) P(V ) ∆ = ∆ is an ideal in P(V ) with P(V ) /∆ = ∆ via π∆ . In particular, spec∆ = \ ∪ ∪ S ∼ ∪ ∪ ∼ V(∆c)= spec P(V ) ∆c . {P ∈ ∪ | ⊆ P} Proof. (1) and (2) follow immediately from Example 2.2.7, and (3) from Example 2.1.10 and Corollary 4.1.19.

~~Proposition 6.4.2. Let ∆ be a simplicial complex on V . The one-to-one correspondences~~

(∆0 ) ∆ V (∆∞) F ∩ ←→ ∪{ } ←→ F ∪ S := G ∆0 F G F G ∆∞ G F =: S F,∩ { ∈ ∩ | ⊆ } ←→ ←→ { ∈ ∪ | ⊆ } F,∪ 0 0 yield binoid isomorphisms (∆ ) = ∆∞ and (∆∞) = ∆ . In particular, F ∩ ∩ ∼ ∪ F ∪ ∩ ∼ ∩

spec∆0 ∼ ∆∞ and spec∆∞ ∼ ∆ ∩ ←→ ∪ \ {∅} ∪ ←→ ∩ := G ∆0 F G F := ∆∞ P(F ) F PF,∩ { ∈ ∩ | 6⊆ } ←→ PF,∪ ∪ \ ←→ are semibinoid isomorphisms, the latter inclusion preserving and the first reversing. In particular, ∅ 0 ∞ ∅ ∞ 0 spec ∆∩ ∼= ∆∪ and spec ∆∪ ∼= ∆∩ as binoids. Proof. By definition, every filter S of ∆0 has to contain the identity element V and since A, B S is ∩ ∈ equivalent to A B S, there is a unique minimal element F S with respect to and all supersets ∩ ∈ ∈ ⊆ of F in ∆ V lie in S. This proves the correspondence S F . Similarly, every filter S of ∆∞ ∪{ } F,∩ ↔ ∪ admits a maximal element F S with respect to and all subsets of F in ∆ V lie in S, which ∈ ⊆ ∪{ } proves the correspondence F SF,∪. By taking complements, we obtain the results for the spectra 0 ↔ of ∆ and ∆∞. Moreover, for two faces F, F ′ ∆ one has ∩ ∪ ∈

~~S S ′ = S ′ and S S ′ = S ′ , F,∩ ∩ F ,∩ F ∪F ,∩ F,∪ ∩ F ,∪ F ∩F ,∪ ∞ and (with respect to the union on ∆∪ )~~

~~′ = ′ and ′ = ′ . PF,∩ ∪PF ,∩ PF ∪F ,∩ PF,∪ ∪PF ,∪ PF ∩F ,∪ 6.4 Topology of simplicial complexes 157~~

0 In particular, S∅,∩ = ∆ V = SV,∪, SV,∩ = V , and S∅,∪ = , which gives V,∩ = ∆ = (∆∩)+ and ∞ ∞∪ { } {∅} P ′ ∅,∪ = ∆∪ = (∆∪ )+. This implies the binoid and semibinoid isomorphisms. Clearly, if F F , P \{∅} ′ ′ ⊆ then F ′,∪ F,∪. On the other hand, if F ′,∪ F,∪, then F F since F F ′,∪. Similarly, P′ ⊆ P P ⊆ P ⊆ ∈ P F F is equivalent to ′ . ⊆ PF,∩ ⊆PF ,∩ Note that for ∆ = P(V ) V the preceding lemma includes all results for P(V ) and P(V ) , cf. \{ } ∩ ∪ Example 1.4.38 and Example 2.2.7. The semibinoid homeomorphisms induced by ι∆ and π∆ are given by the epimorphism

0 ι∗ : specP(V ) spec∆ with J V F J ∆ ∩ −→ ∩ { ⊆ | 6⊆ } 7−→ PF,∩ for = F V , and by the embedding ∅ 6 ⊆ π∗ : spec∆∞ spec P(V ) with P(V ) P(F ) ∆ ∪ −→ ∪ PF,∪ 7−→ \ for F ∆. ∈ Corollary 6.4.3. dim ∆ + 1 = dim ∆0 = dim ∆∞ for every simplicial complex ∆ = P(V ) on V. ∩ ∪ 6 Proof. This is an immediate consequence of Proposition 6.4.2.

0 ∞ The preceding results show that spec ∆∩ and spec∆∪ always have the same cardinality and the 0 ∞ same dimension as topological spaces endowed with the Zariski topology. However, if ∆∩ and ∆∪ are not isomorphic as binoids, the Zariski topologies on their spectra are dual to each other as the following corollary shows.

∞ 0 Corollary 6.4.4. Consider spec∆∪ and spec∆∩ as topological spaces with respect to the Zariski topology. The bijection h : ∆ spec∆∞ −→ ∪ with F , is a homeomorphism if the poset (∆, ) is endowed with the upper topology, and the 7→ PF,∪ ⊆ bijection 0 g : ∆ spec∆ −→ ∩ with F , is a homeomorphism if the poset (∆, ) is endowed with the lower topology 7→ PF,∩ ⊆ ∞ 0 Proof. Since spec ∆∪ and spec∆∩ are ﬁnite sets, the Zariski topology coincides with the lower topology (with respect to ) on them, cf. Remark 4.1.6. Hence, the basic open sets of spec∆∞ are given ⊆ ∪ by and ∅ D = spec∆∞ P {Q ∈ ∪ |Q⊆P} for spec∆∞. The basic open sets in the upper topology of the poset (∆, ) are given by and P ∈ ∪ ⊆ ∅ D = G ∆ F G F { ∈ | ⊆ } for F ∆. This shows that h−1(D ) = D because G F is equivalent to by ∈ PF,∪ F ⊆ PF,∪ ⊆ PG,∪ Proposition 6.4.2. Similarly, g is a homeomorphism if (∆, ) is endowed with the lower topology, ⊆ where the basic open subsets are given by

D = G ∆ G F F { ∈ | ⊆ } for F ∆. By Proposition 6.4.2, G F is equivalent to , and therefore g−1(D )= D , ∈ ⊆ PG,∩ ⊆PF,∩ PF,∩ F where D = spec∆0 for spec∆0 are the basic open sets of spec∆0 . P {Q ∈ ∩ |Q⊆P} P ∈ ∩ ∩ 158 6 Binoids deﬁned by a simplicial complex and simplicial binoids

~~6.5 Simplicial binoids~~

Now we deﬁne the binoid associated to a simplicial complex ∆ such that its binoid algebra is the Stanley-Reisner algebra of ∆. Conversely, we characterize those binoids that yield Stanley-Reisner algebras, cf. Theorem 6.5.8. Then the tools and notations developed in the last sections will be applied to these binoids.

~~Deﬁnition 6.5.1. Let ∆ be a simplicial complex on V . The binoid associated to ∆ is deﬁned to be~~

M∆ := FC(V )/ ∆ , I where ∆ denotes the ideal f FC(V ) supp(f) ∆ in the free commutative binoid FC(V ). A I { ∈ | 6⊆ } binoid M is called a s i m p l i c i a l b i n o i d if M ∼= M∆ for a simplicial complex ∆. In this case, the simplicial complex ∆ is unique and called the u n d e r l y i n g s i m p l i c i a l complex of M. Simplicial binoids together with binoid homomorphisms form a full subcategory of the category comB of commutative binoids, which will be denoted by simplB.

~~Equivalently, one may deﬁne the binoid M∆ by~~

~~M∆ := ϕ : V N map supp ϕ ∆ , { → | ∈ }∪{∞} where supp ϕ = v V ϕ(v) =0 , with addition deﬁned by { ∈ | 6 } ϕ + ψ , if supp(ϕ + ψ) ∆ , ϕ + ψ := ∈ ( , otherwise, ∞ or by~~

~~M∆ := n v F ∆,n N v ∈ v ∈ ∪ {∞} n v∈F o X with addition deﬁned by~~

v∈F ∪G(nv + mv)v , if F G ∆ , nvv + mvv := ∪ ∈ v∈F v∈G (P , otherwise, X X ∞ where n = 0 if v F and m = 0 if v G. In particular, for every f M • there is a unique face v 6∈ v 6∈ ∈ ∆ F ∆ such that f = n v with n 1 for all v F . ∈ v∈F v v ≥ ∈ n ∞ P(PV ) FC Example 6.5.2. M coincides with the free binoid (V ) ∼= (N ) .

Example 6.5.3. If ∆ = , v v V , then M∆ = : FC( v ). In general, if the facets of a {∅ { } | ∈ } v∈V { } simplicial complex ∆ are pairwise disjoint, then M∆ = : FC(F ). See also Corollary 6.5.20 SF facet of ∆ below. S Now we recall the deﬁnition of the Stanley-Reisner algebra of a simplicial complex, cf. [12, Chapter 5.1] or [56, Chapter 5.3].

Deﬁnition 6.5.4. Let ∆ be a simplicial complex on V = v ,...,v and R[X ,...,X ] the { 1 n} 1 n polynomial algebra over the ring R. The Stanley-Reisner ideal I∆ of ∆ is the ideal in R[X ,...,X ] generated by all monomials X X such that v ,...,v ∆. Then the 1 n j1 ··· js { j1 js } 6∈ Stanley-Reisner algebra (alsocalled face ring)of∆over R is given by the graded R - algebra

~~R[X1,...,Xn]/I∆ =: R[∆].~~

~~Example 6.5.5. If ∆=P(V ), then I = (0), hence R[∆] is the polynomial algebra R[X v V ]. ∆ v | ∈ 6.5 Simplicial binoids 159~~

Remark 6.5.6. By deﬁnition, I∆ is generated by squarefree monomials. On the other hand, for every ideal I in R[X1,...,Xn] that is generated by squarefree monomials, there is a simplicial complex on ′ V , #V = n, such that R[X1,...,Xn]/I ∼= R[∆]. Moreover, if ∆ and ∆ are simplicial complexes on ′ the same vertex set V , then ∆ ∆ if and only if I ′ I . ⊆ ∆ ⊆ ∆ As announced, every Stanley-Reisner algebra can be realized as a binoid algebra.

~~Lemma 6.5.7. If ∆ is a simplicial complex and R a ring, then R[∆] ∼= R[M∆ ].~~

Proof. If ∆ be a simplicial complex on V = 1,...,n , then R[M∆ ] = R[FC(V )]/R[ ] with { } I∆ R[ ] = (T a a ) = (Xa a FC(V ), supp a ∆) =: I , I∆ | ∈ I∆ | ∈ 6⊆ where Xa = Xa1 Xan for a = a i. Since Xa I if and only if Xa˜ I fora ˜ = i, 1 ··· n i∈V i ∈ ∈ i∈supp(a) we get a P P I = X a = i,J ∆ = (X X j ,...,j ∆) = I∆ . 6⊆ j1 ··· js |{ 1 s} 6⊆ i∈J⊆V X Thus, R[M∆ ] is the Stanley-Reisner algebra of ∆.

~~Simplicial binoids can be characterized as follows.~~

Theorem 6.5.8. Let ∆ be a simplicial complex on V . The binoid M∆ is ﬁnitely generated by #V = n elements, semifree, and reduced. Conversely, every commutative binoid M satisfying these properties is a simplicial binoid. More precisely, M is isomorphic to M∆ , where ∆ := F W w = { ⊆ | w∈F 6 ∞} for a minimal generating set W of M. P

Proof. By Corollary 2.1.21 and Lemma 1.4.2, we only need to show that M∆ is reduced. This follows immediately from the fact that supp(f + f) = supp(f) supp(f) = supp(f). For the converse note ∪ that such a generating set exists by Proposition 1.4.27. So take a semibasis W of M and consider the canonical binoid epimorphism ϕ : FC(W ) M with w w. By Lemma 1.4.23, M is cancellative. −→ 7→ Therefore, if f,g FC(W ) with f = n w and g = m w, then ϕ(f)= ϕ(g) is equivalent ∈ w∈W w w∈W w to ϕ(f)= ϕ(g)= or n = m for all w W , and hence f = g. Furthermore, we have by Lemma ∞ w w P ∈ P 1.4.2 that f + f + g = implies that f + g = for all f,g M. Applying this successively to ∞ ∞ ∈ n w = , where T W , n 1, w T , yields w = because if u T with n 2, w∈T w ∞ ⊆ w ≥ ∈ w∈T ∞ ∈ u ≥ then P P = n w = u + u + (n 2)u + n w ∞ w u − w wX∈T TX\{u} = u + (n 2)u + n w u − w TX\{u} = ...

= u + nww . TX\{u} Thus, we have shown that = , where = w FC(W ) w = in M , which ∼ϕ ∼I I { w∈T ∈ | w∈T ∞ } gives FC(W )/ = M by Proposition 1.6.2. Since I ∼ P P = w FC(W ) F ∆ , I ∈ 6⊆ n w∈F o X where ∆ := F W w = is a simplicial complex on W , we obtain = . Thus, { ⊆ | w∈F 6 ∞} I I∆ M = FC(W )/ ∆ . ∼ I P 160 6 Binoids deﬁned by a simplicial complex and simplicial binoids

~~Example 6.5.9. The binoid FC(x, y)/(2x + y = ) is not a simplicial binoid since it is not (strongly) ∞ reduced, cf. Lemma 1.4.2(2).~~

Corollary 6.5.10. Let be a radical ideal in M. If M is a simplicial binoid, then so is M/ . I I Proof. By the characterization of simplicial binoids given in Theorem 6.5.8 above, the statement follows from Lemma 2.1.19 and Lemma 2.3.8, and the simple fact that f + f + g = implies ∞ f + g = also holds in M/ . ∞ I

Example 6.5.11. Let ∆ be a simplicial complex on V and spec M∆ . By Corollary 6.5.13 below, ˜ ˜ P ∈ = F˜ = v V v F for some face F ∆. If A = F ∆ F facet so that ∆ = F ∈A P(F ), P P h ∈ | 6∈ i ˜ ∈ ˜ { ∈ | } then M∆ / = M∆˜ , where ∆= P(F F ). PF F ∈A ∩ S ∞ Lemma 6.5.12. The booleanizationS of M∆ is isomorphic to ∆∪ .

∞ Proof. By Proposition 4.2.2, the binoid homomorphism ϕ : M∆ ∆∪ with v∈F nvv F , where → ∞ 7→ F ∆ and n 1, induces a binoid homomorphismϕ ¯ : (M∆ ) ∆ with v F , where ∈ v ≥ bool → ∪ P v∈F 7→ F ∆, which is obviously bijective. ∈ P

Corollary 6.5.13. Let ∆ be a simplicial complex on V . Every face F ∆ deﬁnes a ﬁlter in M∆ , ∈ namely Filt(F ) = f M∆ supp(f) F , and hence a prime ideal M∆ Filt(F ) such that the { ∈ | ⊆ } \ canonical maps

~~∼ ∼ (M∆ ) M∆ ∆ spec M∆ F \{ } ←→ ∩ ←→ S v V v S 7−→ { ∈ | ∈ } Filt(F ) [ F v V v F ←− 7−→ h ∈ | 6∈ i v V v [ { ∈ | 6∈ P} ←− P~~

0 ∅ are semibinoid isomorphism. In particular, the binoids ∆ , (M∆ ) , and spec M∆ are isomorphic. ∩ F ∩ Proof. Since Filt(G G′) = Filt(G) Filt(G′) and Filt( )= M ×, the canonical maps on the left-hand ∩ ∩ ∅ ∆ side are semibinoid homomorphism, which are inverse to each other. The rest follows from the ﬁrst part and the following identiﬁcations

~~∞ spec M∆ = spec (M∆ ) = spec∆ = ∆ , ∼ B ∼ ∪ ∼ ∩ which are due to Corollary 4.2.7, Lemma 6.5.12, and Proposition 6.4.2.~~

Corollary 6.5.14. Let ∆ be a simplicial complex on V and spec M∆ , F ∆. Then PF ∈ ∈ (1) dim = dim F +1. In particular, dim M∆ = dim ∆ + 1. PF (2) F (M∆ )= f (∆) for all i 0, 1,...,d , d := dim M∆ ; that is, F (M )= f(∆). i i−1 ∈{ } ∆ (3) The minimal prime ideals of M∆ are of the form = f M∆ supp(f) V F for some PF { ∈ | ⊆ \ } facet F ∆; that is, the irreducible components of spec M∆ correspond to the facets of ∆. ∈

Proof. By Corollary 6.5.13, every prime ideal in spec M∆ is of the form := v v V F for PF h | ∈ \ i some F ∆. In particular, if dim F = r 1, say F = v ,...,v , then ∈ − { 1 r} = ( ( ( ( , PF Pr Pr−1 ··· P1 P0 where = v V v v ,...,v , i 1,...,r , and = v v V = (M∆ ) . Hence, every Pi h ∈ | 6∈ { 1 i}i ∈ { } P0 h | ∈ i + strictly increasing chain of faces is equivalent to a strictly decreasing chain of prime ideals and vice versa. This implies all statements, where the supplement of (3) is due to Corollary 4.1.17. 6.5 Simplicial binoids 161

Remark 6.5.15. Let ∆ be a simplicial complex on V = 1,...,n and K a ﬁeld. Now we can easily { } visualize the K - spectrum of a simplicial binoid M∆ for an arbitrary simplicial complex ∆ on V . By Proposition 4.4.7, we have

K-spec M∆ = V (K[ ]) , ∼ K PF F ∈∆[ facet where = v V v F by Corollary 6.5.14(3), and PF h ∈ | 6∈ i V (K[ ]) = (a ,...,a ) An(K) a = 0 if i F and a K otherwise K PF ∼ { 1 n ∈ | i 6∈ i ∈ } by Remark 4.4.4. Hence,

n K-spec M∆ = A(F ) A (K) = K-spec MP(V ) , ∼ ⊆ ∼ F ∈∆ facet [ where A(F )= A A with 1 ×···× n A1(K) , if i F, Ai := ∈ ( 0 , otherwise, { } In particular, every union of coordinate planes is realizable as the K - spectrum of a simplicial binoid, and therefore (see below) as the K - spectrum of its underlying simplicial complex. For instance, if n = 3, then

~~R- spec MP(V ) V \~~

In topology, the geometric realization of a simplicial complex ∆ on V = 1,...,n is given by { } ϕ R-spec∆ ϕ = (a ,...,a ) An(R),a 0,a + + a =1 , { ∈ | 1 n ∈ i ≥ 1 ··· n } cf. [25, Chapter II.2.1]. b

~~Proposition 6.5.16. Let ∆ be a simplicial complex on V and N a commutative binoid. Every~~

N - point ρ : V N of ∆ factors uniquely through M∆ ; that is, there is a unique binoid homomorphism → ρ˜ : M∆ N such that ρι˜ = ρ, where ι : V M∆ denotes the canonical map v v. In particular, → → 7→

~~N- spec M∆ ∼= N- spec∆ as semibinoids and R- Spec R[∆] ∼= R- spec∆ for a ring R. 162 6 Binoids deﬁned by a simplicial complex and simplicial binoids~~

Proof. Since every element f M • can be written uniquely as f = n v with n 1, v F , ∈ ∆ v∈F v v ≥ ∈ for some F ∆, the mapρ ˜ : M∆ N withρ ˜(f) := n ρ(v) andρ ˜( ) := is well-defined ∈ → v∈F v P ∞ ∞ and satisfiesρι ˜ = ρ by definition. To show thatρ ˜ is a binoid homomorphism let f = n v and P v∈F v g = m v with F, G ∆ such that n ,m 1. Then v∈g v ∈ v v ≥ P P ρ˜(f)+˜ρ(g) = nvρ(v)+ mvρ(v) = (nv + mv)ρ(v) vX∈F vX∈F v∈XF ∪G in N, where n := 0 if v F and m := 0 if v G. In particular, n + m 1 0 for all v F G. v 6∈ v 6∈ v v − ≥ ∈ ∪ Thus, if F G is not a face in ∆, then f + g = in M∆ and ∪ ∞ ρ˜(f)+˜ρ(g) = ρ(v) + (n + m 1)ρ(v) v v − v∈XF ∪G v∈XF ∪G = + (n + m 1)ρ(v) ∞ v v − v∈XF ∪G = ∞ since ρ is an N -point. Hence,ρ ˜(f)+˜ρ(g)= =ρ ˜( )=˜ρ(f + g). If, on the other hand, F G ∆, ∞ ∞ ∪ ∈ then f +g = (n +m )v M∆ , and thereforeρ ˜(f +g)= (n +m )ρ(v)=˜ρ(f)+˜ρ(g) v∈F ∪G n v ∈ v∈F ∪G n v as above. For the supplement consider the canonical map P P

ψ : N-spec M∆ N- spec ∆ with ϕ ϕι =: ϕ , −→ 7−→ |V where ι : V M∆ , v v, as in the proposition. The bijectivity of ψ follows from the ﬁrst part. → 7→ Obviously, ψ is a semigroup homomorphism which satisﬁes χ ε, where χ is the absorbing {0} 7→ {0} element of N-spec M∆ (because M∆ is positive) and ε : v , v V , that of N-spec∆. The 7→ ∞ ∈ statement on the R - points of the Stanley-Reisner algebra R[∆] and ∆ follows from Proposition 3.2.3 and Lemma 6.5.7.

˜ ∞ ∞ Proposition 6.5.17. Every binoid homomorphism ϕ : ∆∪ ∆∪ gives rise to a homomorphism ˜ → φ : M∆˜ M∆ of the simplicial binoids deﬁned by ∆ and ∆. This binoid homomorphism commutes → ∞ via (M∆ ) = ∆ with ϕ. Hence, ϕ = (φ) and there is a commutative diagram bool ∼ ∪ B

~~φ M∆˜ / M∆ ,~~

πbool πbool B(φ) ˜ ∞ ∞ ∆∪ / ∆∪ where π is the canonical binoid homomorphism n v F . Moreover, (φ) is surjective if bool v∈F v 7→ B and only if φ is so. P Proof. Let ϕ : ∆˜ ∞ ∆∞ be a binoid homomorphism. For f = n v M • with n 1 and ∪ → ∪ v∈F v ∈ ∆˜ v ≥ F ∆˜ deﬁne ∈ P

nv w , if ϕ(F ) ∆ , φ(f) := v∈F w∈ϕ({v}) ∈ ( , otherwise (i.e. when ϕ(F )= in ∆∞) . ∞P P ∞ ∪ If g M • with g = m v, m 1, G ∆,˜ then ∈ ∆˜ v∈G v v ≥ ∈ P f + g = nvv + mvv = (nv + mv)v , vX∈F vX∈G v∈XF ∪G 6.5 Simplicial binoids 163 where n := 0 if v F and m := 0 if v G. Thus, for f + g = v 6∈ v 6∈ 6 ∞

φ(f + g) = (nv + mv) w v∈XF ∪G w∈Xϕ(v) = nv w + mv w v∈XF ∪G w∈Xϕ(v) v∈XF ∪G w∈Xϕ(v) = nv w + mv w = φ(f)+ φ(g) . vX∈F w∈Xϕ(v) vX∈G w∈Xϕ(v) On the other hand, f + g = means F G ∆˜ (or equivalently F G = in ∆˜ ∞), which implies ∞ ∪ 6∈ ∪ ∞ ∪ that ϕ(F G) = since ϕ is a binoid homomorphism. Hence, φ(f + g) = . By Corollary 4.2.9 ∪ ∞ ∞ and Lemma 6.5.12, every binoid homomorphism φ : M∆˜ M∆ induces such a commutative diagram → with (φ)( v )= π φ(v). Thus, ϕ = (φ) by the ﬁrst part. The supplement is clear. B { } bool B Recall from Section 6.2, cf. Lemma 6.2.7, that we have the following binoid homomorphisms:

, if F = , ∅ ∅ ∞ ˜ ∞ −1 ϕλ : ∆∪ ∆∪ with F  , if F = or λ (v)= for some v F, −→ 7−→ ∞ ∞ ∅ ∈ λ−1(F ) , otherwise,  if λ : (V,˜ ∆)˜ (V, ∆) is simplicial.  → , if F = or λ(F ) ∆ , α ˜ ∞ ∞ ϕλ : ∆∪ ∆∪ with F ∞ ∞ 6∈ −→ 7−→ (λ(F ) , otherwise, if λ : (V,˜ ∆)˜ (V, ∆) is α-simplicial. → , if F = or λ−1(F ) ∆˜ , β ∞ ˜ ∞ ϕλ : ∆∪ ∆∪ with F ∞ ∞ 6∈ −→ 7−→ (λ−1(F ) , otherwise, if λ : (V,˜ ∆)˜ (V, ∆) is β-simplicial. → Corollary 6.5.18. (1) Every simplicial morphism λ : (V,˜ ∆)˜ (V, ∆) induces a binoid homomorphism → , if λ−1(v)= for some v F, φ : M∆ M∆˜ , n v ∞ ∅ ∈ λ −→ v 7−→ ( w∈λ−1(F ) nλ(w)w , otherwise, vX∈F P that is, ∆ M∆ is a contravariant functor SC simplB. If λ is injective, then φ is surjective. 7→ → λ (2) Every α-simplicial morphism λ : (V,˜ ∆)˜ (V, ∆) induces a binoid homomorphism → α φ : M∆˜ M∆ , n v n λ(v) , λ −→ v 7−→ v vX∈F vX∈F α α that is, ∆ M∆ is a covariant functor SC simplB. If λ is surjective, then so is φ . 7→ → λ (3) Every β-simplicial morphism λ : (V,˜ ∆)˜ (V, ∆) induces a binoid homomorphism → β φλ : M∆ M∆˜ , nvv nλ(w)w , −→ 7−→ −1 vX∈F w∈λX(F )

~~β β that is, ∆ M∆ is a contravariant functor SC simplB. If λ is bijective, then φ is surjective. 7→ → λ 164 6 Binoids deﬁned by a simplicial complex and simplicial binoids~~

~~In particular, there are three commuting diagrams~~

~~α β φλ φλ φλ M∆ / M∆˜ M∆˜ / M∆ M∆ / M∆˜ ,~~

~~πbool πbool πbool πbool πbool πbool α β ϕλ ϕ ϕ ∞ ˜ ∞ ˜ ∞ λ ∞ ∞ λ ˜ ∞ ∆∪ / ∆∪ ∆∪ / ∆∪ ∆∪ / ∆∪~~

α β where ϕλ, ϕλ , and ϕλ are the binoid homomorphisms from above and πbool the canonical binoid homomorphism n v F . v∈F v 7→ P α β Proof. All results follow from Proposition 6.5.17 using the binoid homomorphisms ϕλ, ϕλ , and ϕλ α β from above to obtain φλ, φλ , and φλ, respectively.

Remark 6.5.19. The preceding corollary combined with Proposition 4.4.2 yields the following homeomorphisms between K - spectra. (1) If λ : (V,˜ ∆)˜ (V, ∆) is simplicial, then → ∗ φλ K-spec M∆ K-spec M∆˜ with ϕ ϕ˜ , −→ 7−→ where , if λ−1(v)= for some v F, ϕ˜ nvv = ∞ ∅ ∈ nλ(w) ( w∈λ−1(F ) ϕ(w) , otherwise. vX∈F (2) If λ : (V,˜ ∆)˜ (V, ∆) is α-simplicial,Q then → (φα)∗ λ nv K-spec M∆˜ K-spec M∆ with ϕ ϕ˜ : n v ϕ(λ(v)) . −→ 7−→ v 7−→ vX∈F vY∈F (3) If λ : (V,˜ ∆)˜ (V, ∆) is β-simplicial, then →

(φβ )∗ λ nλ(w) K-spec M∆ K-spec M∆˜ with ϕ ϕ˜ : nvv ϕ(w) . −→ 7−→ 7−→ −1 vX∈F w∈λY(F ) Corollary 6.5.20. If ∆ is a simplicial complex on V , i I, I ﬁnite, then i i ∈

~~MQ ∆ M∆ and MU ∆ : M∆ . i∈I i ∼= i i∈I i ∼= i i^∈I i[∈I In particular, if N is a commutative binoid, then~~

~~N- spec MQ ∆ N- spec M∆ , i∈I i ∼= i Yi∈I and if N an integral binoid, then~~

N- spec MU ∆ = N- spec M∆i . i∈I i ∼ · i[∈I Proof. The α-simplicial morphisms (V , ∆ ) ֒ ( V , ∆ ) with v (v; k), k I, cf. Example k k → i∈I i i∈I i 7→ ∈ 6.2.13, induce by Corollary 6.5.18(2) binoid homomorphisms ι : M∆ MQ ∆ , k I, which give U Q k k → i∈I i ∈ rise to the binoid homomorphism M∆ MQ ∆ with f ι (f ) by Proposition i∈I i → i∈I i ∧i∈I i 7→ i∈I i i 1.8.10. Since every h MQi∈I ∆i can be written as h = nvιi(v) for unique faces Fi ∆i ∈ V i∈I v∈Fi P ∈ P P 6.5 Simplicial binoids 165 such that n 1 for all v F , i I, this binoid homomorphism is an isomorphism. Similarly, the v ≥ ∈ i ∈ ,α-simplicial morphisms (V , ∆ ) ֒ ( V , ∆ ) with v (v; k), k I, cf. Example 6.2.13 k k → i∈I i i∈I i 7→ ∈ induce by Corollary 6.5.18(2) binoid homomorphisms M∆ MU ∆ , k I, which give rise to U U k → i∈I i ∈ the binoid homomorphism : M∆ MU ∆ with (f; k) f by Proposition 1.9.9. Since every i∈I i → i∈I i 7→ h MU ∆ can be written as h = n v for a unique face F ∆ , i I, such that n 1 for ∈ i∈I i S v∈F v ∈ i ∈ v ≥ all v F , this binoid homomorphism is also an isomorphism. The isomorphisms of the supplement ∈ P follow from Proposition 1.8.12 and Corollary 1.9.10.

Remark 6.5.21. In combinatorial commutative algebra, the product of two simplicial complexes ∆1 and ∆ is called the jo i n and is usually denoted by ∆ ∆ , cf. [56, Deﬁnition 5.2.4]. It is known that 2 1 ∗ 2 the Stanley-Reisner algebra K[∆ ∆ ] of the join is given by the tensor product K[∆ ] K[∆ ], 1 ∗ 2 1 ⊗K 2 with which one proves, for instance, that ∆ ∆ is Cohen-Macaulay if and only if ∆ and ∆ 1 ∗ 2 1 2 are Cohen-Macaulay, cf. [56, Proposition 5.3.16]. The result for the tensor product can be deduced elegantly from the theory developed in this chapter. By Lemma 6.5.7 and Corollary 6.5.20, we have

K[∆] = K[M∆ ] and MQi∈I ∆i = i∈I M∆ . Hence, V K[∆ ∆ ] = K[M∆ ×∆ ] = K[M∆ M∆ ] = K[M∆ ] K[M∆ ] = K[∆ ] K[∆ ] , 1 × 2 1 2 1 ∧ 2 1 ⊗K 2 1 ⊗K 2 where the third equality is due to Corollary 3.5.2. Example 6.5.22. Let ∆ and ∆˜ be simplicial complexes on V such that ∆˜ ∆. The morphisms ⊆ (V, ∆) id (V,˜ ∆)˜ id (V, ∆) , −→ −→ where the ﬁrst morphism is α-simplicial and the latter simplicial and β-simplicial, cf. Example 6.2.11, induce the same binoid epimorphism

~~M∆ M∆˜ with n v n v . −→ v 7−→ v vX∈F vX∈F~~

~~In particular, for every simplicial complex ∆ on V there is a binoid epimorphism MP(V ) M∆ , which → yields the canonical K - algebra epimomorphism~~

~~K[P(V )] = K[X v V ] K[X v V ]/I∆ = K[M ] = K[∆] , v | ∈ −→ v | ∈ ∆~~

. ( K a ring. Note that if ∆ ( P(V ) there is no binoid embedding M∆ ֒ MP(V → Example 6.5.23. Let ∆ be a simplicial complex on V , T V , and ∆(T )= F ∆ F T . The ⊆ { ∈ | ⊆ } simplicial and α-simplicial morphism, cf. Example 6.2.12,

~~λ : (T, ∆(T )) (V, ∆) with v v → 7−→ induces the binoid homomorphisms~~

α φλ φλ M∆(T ) M∆ M∆(T ) −→ −→ with nvv , if F ∆(T ) , n v n v v∈F ∈ v 7−→ v 7−→ v∈F v∈F (P , otherwise. X X ∞ α α φλ is a binoid epimorphism and φλ a binoid embedding, which satisfy φλ φλ = idM∆(T ) . We consider ◦ r ∞ two special cases. If T = F ∆ with #F = r, then ∆(T )=P(F ) and MP(F ) = (N ) , which gives ∈ ∼ α r ∞ φλ φλ r ∞ (N ) M∆ (N ) , −→ −→ 166 6 Binoids deﬁned by a simplicial complex and simplicial binoids

~~where φ is the canonical projection M∆ M∆ / with = v V v F spec M∆ . By λ → PF PF h ∈ | 6∈ i ∈ Proposition 1.5.6 and Remark 4.4.4, this yields for a ﬁeld K~~

∗ α ∗ r φλ (φλ ) r A (K) K-spec M∆ A (K) . −→ −→ n ∞ The other special case is when ∆ = P(V ). Then M∆ ∼= (N ) , where n = #V . As observed in Example 6.2.12, the morphism λ : (T, ∆(T )) (V, P(V )) is also β-simplicial in this case. Hence, → β n ∞ φ , φ : (N ) M∆(T ) λ λ −→ are the two binoid epimorphisms generated by

~~i , if i T, β i , if i T, φλ(ei)= ∈ and φλ(ei)= ∈ ( , otherwise, (0 , otherwise. ∞ These yield two diﬀerent embeddings~~

~~∗ β ∗ n φ , (φ ) : K-spec M∆(T ) A (K) . λ λ −→ If V = 1, 2 , T = 1 , and K = R, these two embeddings A1(R) ֒ A2(R) are given by { } { } →~~

∗ ∗ via φλ via (φβ) Example 6.5.24. Let I be ﬁnite and ∆ a simplicial complex on V , i I. For every k I, the i i ∈ ∈ simplicial and α-simplicial morphism ι : (V , ∆ ) ( V , ∆ ), (v; k) (v; k), cf. Example k k k → i∈I i i∈I i 7→ 6.2.13, induces the binoid homomorphisms U U (φα )∗ φ∗ ιk ιk M : M∆ M ∆k −→ i −→ ∆k i[∈I with f , if i = k , f (f; k) and (f; i) 7−→ 7−→ ( , otherwise. ∞ α φιk is a binoid epimorphism and φιk a binoid embedding. Moreover, the α-simplicial morphism ˜ι (V , ∆ ) ( V , ∆ ) with (v; k) (v; k), cf. Example 6.2.13, induces a binoid embedding k k k → i∈I i i∈I i 7→

~~U α Q φ : M∆ M∆i with f 0 0 f 0 0 , ιk k −→ 7−→ ∧···∧ ∧ ∧ ∧···∧ i^∈I where f is the kth entry. Index~~

An(K)-connected, 143 free commutative –, 17 absorbing, 11 free of asymptotic torsion –, 22 algebra, 97 generating set of a –, 16 – homomorphism, 97 generators of a –, 16 binoid –, 99 graded –, 140 group –, 97 integral –, 19 monoid –, 97 integral closure of a –, 75 structure homomorphism of an –, 97 integrally closed –, 75 Artin-Rees lemma, 84 minimal generating set of a –, 16 ascending chain condition (a.c.c.) monomial –, 83 – on congruences, 32 N -point of a –, 28 – on ideals, 84 N -spectrum of a –, 28 associated, 34 noetherian –, 32 associated graded noetherian – (ideals), 84 – binoid, 108 normal –, 75 – ring, 108 normalization of a –, 75 asymptotic torsion, 22 positive –, 20 asymptotically equivalent, 22 positively graded –, 140 reduced –, 19 basis, 17 regular –, 22 bidual, 29 semifree –, 18 binoid, 11 semifree up to idempotence –, 146 – algebra, 12, 99 separated –, 135 – generated by, 16 simplicial –, 160 – group, 20 strongly reduced –, 19 – homomorphism, 14 torsion-free –, 21 – integral over another binoid, 75 torsion-free up to nilpotence –, 21 basis of a –, 17 trivial –, 12 bidual of a –, 29 unit group of a –, 20 blowup –, 107 zero –, 12 boolean –, 21 binoid group, 20 booleanization of a –, 119 binoid homomorphism, 14 cancellation – of, 35 local –, 80 cancellative –, 22 bipointed union, 43 difference –, 74 blowup binoid, 107 dimension of a –, 121 boolean, 21 dual of a –, 29 – point, 28 F -vector of a –, 122 booleanization, 119 finite –, 16 finitely generated –, 16 cancellative, 22 free –, 17 – K -component, 126 168 Index

– K -part, 126 associated –s, 34 character, 127 asymptotically equivalent –s, 22 K -–, 127 cancellative –, 22 closure, 116 degree of an –, 140 colimit, 54 functionally equivalent –s, 126 completion homogeneous –, 98 a - adic – (of a ring), 135 idempotent –, 21 – of a binoid, 136 identity–, 11 component integral –, 19 cancellative K -–, 126 inverse –, 20 irreducible –, 117 loopfree –, 58 congruence, 30 nilpotent –, 19 – generated by relations, 31 separated –, 138 a.c.c. on –s, 32 strongly compatible –, 56 finitely generated –, 31 support of an –, 18, 146 ideal –, 33 torsion –, 21 identity –, 30 unseparated –, 138 intersection of –s, 30 weakly (un-) separated –, 138 Rees –, 82 embedding, 14 universal –, 30 epimorphism, 14 connected, 128 An(K)-–, 143 F -vector, 122 f -vector, 145 degree, 140 face, 145 Dickson’s Lemma, 81, 137 – ring, 160 difference binoid, 74 dimension of a –, 145 dimension facet, 145 – of a binoid, 121 filter, 25 – of a face, 145 – generated by a subset, 26 – of a prime ideal, 122 filtrum, 25 – of a simplicial complex, 145 free of asymptotic torsion, 22 combinatorial –, 121 functionally equivalent, 126 separated –, 135 direct limit, 54 generic point, 117 direct sum grading, 140 – of N -sets, 47 positive –, 140 – of binoids, 14 group algebra, 97 – of monoids, 13 – of pointed sets, 13 Hilbert-Samuel function, 24, 82 directed Hilbert-Samuel series, 24 – set, 53 homomorphism – system, 53 anti-indicator –, 27 disconnected, 128 binoid –, 14 disjoint union, 40 image of a –, 14 – of simplicial complexes, 148 indicator –, 27 dual, 29 integral –, 75 kernel of a –, 14 edge, 145 N - binoid –, 49 element N - multi binoid –, 51 absorbing –, 11 semibinoid –, 14 Index 169 ideal, 79 localization, 70 – generated by, 81 loop, 58 a.c.c. on –s, 84 –free, 58 extended –, 80 length of a –, 58 finitely generated –, 81 lower topology, 114 generators of an –, 81 minimal set of –, 81 map maximal –, 80 N -–, 46 minimal prime of an –, 93 pointed, 13 monomial –, 104 maximal ideal, 80 prime –, 85 minimal prime ideal, 93 principal –, 81 monoid, 11 radical –, 94 – algebra, 97 radical of an –, 94 affine –, 76 Rees quotient by an –, 82 boolean –, 21 separating –, 135 (K-) character of a –, 127 Stanley-Reisner –, 160 cancellative –, 22 unit –, 79 positive –, 20 idempotent, 21 separative –, 128 image, 14 unit group of a –, 20 indemcomposable injective, 106 monomial ideal, 104 indicator function, 27 monomorphism, 14 anti- –, 27 N - binoid, 49 inductive limit, 54 – homomorphism, 49 initial pair, 58 finitely generated –, 49 injective hull, 106 structure homomorphism of an –, 49 integral, 19 N -map, 46 – closure, 75 N - multi – homomorphism, 75 – binoid homomorphism, 51 – over, 75 – map, 48 integrally closed, 75 N - point inverse, 20 – of a binoid, 28 – limit, 55 – of a simplicial complex, 155 irreducible, 20, 117 N -set, 46 – component, 117 finitely generated –, 46 isomorphism, 14 N - spectrum – of a binoid, 28 K - character, see monoid – of a simplicial complex, 155 K - point, see N - point Nakayama’s lemma, 85 K -spectrum, 123 nil radical, 94 Krull’s intersection theorem, 84 nilpotent, 19 Krull’s principal ideal theorem, 123 nonface, 145 Krull’s theorem, 85 normal, 75 normalization, 75 lattice, 80 complete –, 80 operation, 46 limit, 55 order function, 139 direct/inductive –, 54 positive –, 139 inverse/projective –, 55 strong projective –, 56 partition 170 Index

– of a set, 148 boolean –, 21 path algebra, 103 cancellative –, 22 point finite –, 16 K -–, see N - point finitely generated –, 16 N -–, 28 generating set of a –, 16 boolean –, 28 generators of a –, 16 characteristic –, 28 integral –, 19 special –, 28 reduced –, 19 pointed set, 13 regular –, 22 operation on a –, 46 semifree –, 18 pointed union semifree up to idempotence –, 146 – of N -sets, 47 strongly reduced –, 19 – of binoids, 42 torsion-free –, 21 – of pointed sets, 42 torsion-free up to nilpotence –, 21 positive, 20 semifree, 18 prime ideal, 85 – up to idempotence, 146 dimension of a –, 122 semigroup, 11 height of a –, 122 boolean –, 21 minimal –, 93 semilattice, 21 product set – of N -sets, 47 partition of a –, 148 – of binoids, 14 simplicial – of monoids, 13 binoid, 160 – of pointed sets, 13 simplicial complex, 145 – of simplicial complexes, 148 connected –, 148 smash –, 37 dimension of a –, 145 subdirect –, 95 disjoint union of –s, 148 product order, 81 edge of a –, 145 projective limit, 55 f -vector of a –, 145 strong –, 56 face of a –, 145 facet of a –, 145 quiver, 103 join of –s, 167 – with 1, 103 N -point of a –, 155 radical N -spectrum of a –, 155 – ideal, 94 nonface of a –, 145 – of an ideal, 94 product of –s, 148 nil –, 94 pure –, 145 reduced, 19 underlying –, 160 strongly –, 19 vertice of a –, 145 regular, 22 simplicial morphism, 150 relation, 31 α - –, 150 binomial –, 57 β - –, 150 monomial –, 57 smash product, 37 unmixed –, 57 – of N - binoids, 47 – of N -sets, 47 semibasis, 18, 146 space semibinoid, 11 connected –, 128 – generated by, 16 disconnected –, 128 – homomorphism, 14 irreducible –, 20, 117 Index 171 spectrum, 85 – of a ring, 123 extended –, 113 K -–, 123 N - – of a binoid, 28 N - – of a simplicial complex, 155 Stanley-Reisner algebra, 160 Stanley-Reisner ideal, 160 strong projective limit, 56 stronlgy compatible, 56 structure homomorphism – of an algebra, 97 – of an N - binoid, 49 subbinoid, 11 submonoid, 11 subsemibinoid, 11 subset-closed, 25 superset-closed, 25 support, 18, 146 system directed –, 53 inverse –, 55 topology lower –, 114 upper –, 115 Zariski –, 114, 124 torsion-free, 21 – up to nilpotence, 21 ultrafilter, 93 unit, 20 – group, 20 upper topology, 115 vertice – of a simplicial complex, 145

~~Zariski topology – on K-Spec K[M], 124 – on spec M, 114~~

~~Symbols~~

congruence on a binoid, page 118 ∼bool congruence on a binoid M, S M a submonoid, page 35 ∼can,S ⊆ congruence on a binoid, page 35 ∼ae congruence on a binoid, page 126 ∼fe identity congruence, page 30 ∼id congruence on a binoid, page 34 ∼int congruence on a binoid, page 33 ∼ker ϕ congruence on a binoid, page 34 ∼pos congruence on a binoid, page 34 ∼red congruence on a binoid M generated by relations R M M, page 31 ∼R ⊆ × congruence on a binoid, page 35 ∼tf universal congruence, page 30 ∼u congruence on a binoid with respect to ϕ, page 30 ∼ϕ equivalence relation on the disjoint union of pointed sets, page 42 ∼∞ equivalence relation on the disjoint union of positive binoids, page 42 ∼• relation on the product of pointed sets, page 37 ∼∧

∧ congruence on the smash product of a finite family of N -sets, page 47 ∼ N ∆ semibinoid defined by ∆ with respect to , page 146 ∩ ∩ ∆∞ binoid defined by ∆ with respect to , page 146 ∪ ∪ A subbinoid generated by the subset A M, page 15 h i ⊆ V ideal in M generated by V M, page 81 M h i ⊆ α = χ , anti-indicator function with respect to the subset N M, page 27 N M\N ⊆

~~χN indicator function with respect to the subset N, page 27~~

~~ϕ( )+ N extended ideal of M with respect to ϕ : M N, page 80 I I ⊆ →~~

~~ϕλ binoid homomorphism induced by a simplicial morphism λ, page 151~~

~~α ϕλ binoid homomorphism induced by an α-simplicial morphism λ, page 151 β ϕλ binoid homomorphism induced by a β-simplicial morphism λ, page 152 174 Index~~

i∈I Si disjoint union of a family of sets, page 40 U i∈I ∆i disjoint union of a family of simplicial complexes, page 148 U S pointed union of a family of pointed sets, page 42 · i∈I i S: i∈I Mi bipointed union of a family of positive binoids, page 43 S i∈I Mi product of a family of binoids, page 14 Q i∈I Mi product of a family of monoids, page 13 Q i∈I Si product of a family of pointed sets, page 13 Q i∈I ∆i product of a family of simplicial complexes, page 148 Q i∈I Mi smash product of a family of binoids, page 37 V i∈I Si smash product of a family of pointed sets, page 37 V N,i∈I Mi smash product over N of a ﬁnite family of N - binoids, page 50 V N,i∈I Si smash product over N of a ﬁnite family of N -sets, page 47 V i∈I Mi direct sum of a family binoids, page 14 L i∈I Mi direct sum of a family monoids, page 13 L i∈I Si direct sum of a family of pointed sets, page 13

~~ψLλ binoid homomorphism induced by a simplicial morphism λ, page 152~~

~~α ψλ binoid homomorphism induced by an α-simplicial morphism λ, page 153 √ radical of , page 93 I I B category of binoids, page 15~~

~~BN category of N - binoids, page 49 boolB category of commutative boolean binoids, page 120~~

(M) = D(f) f M , page 119 B { | ∈ } (M) = ( (M), , spec M, ), page 119 B ∩ B ∩ ∅ bool(M) set of all idempotent elements in M, page 21 can(M) set of all cancellative elements in M, page 22 comB category of commutative binoids, page 15 const(F ) constant term of F K[M], page 101 ∈ D( ) opensetintheZariskitopologyonspec M, page 113 − D ( ) open set in the Zariski topology on K-spec M, page 124 K − D∅( ) opensetintheZariskitopologyonspec∅ M, page 113 − diﬀ(M) diﬀerence binoid of the binoid M, page 74 Symbols 175 dim M dimension of the binoid M, page 121 dim dimension of the prime ideal , page 122 P P dimsep M separated dimension of the binoid M, page 135 dim F dimension of the face F , page 145 dim ∆ dimension of the simplicial complex ∆, page 145 dimK V dimension of the K - module V , page 104

Dim R dimension of the ring R, page 138 ei = (0,..., 0, 1, 0,..., 0), where 1 is the ith entry, page 16 aei = (0,..., 0, a, 0,..., 0), where a is the ith entry, page 16 e = (0,..., 0, , 0,..., 0), where is the ith entry, page 16 i,∞ ∞ ∞ e =0 0 1 0 0, where 1 is the ith entry, page 38 i ∧···∧ ∧ ∧ ∧···∧ e =0 0 1 0 0, where 1 is the ith entry, page 50 bi,N ∧N ···∧N ∧N ∧N ∧N ···∧N ae =0 0 a 0 0 M , where a M is the ith entry, page 38 b i ∧···∧ ∧ ∧ ∧···∧ ∈ k∈I k ∈ i ae =0 0 a 0 V 0, where a is the ith entry, page 50 bi,N ∧N ···∧N ∧N ∧N ∧N ···∧N End V ring of all K - module endomorphisms of V , page 106 b K f (∆) = # F ∆ dim F = i , page 145 i { ∈ | } f(∆) = (f−1(∆),f0(∆),...,fd(∆)), f -vector of ∆, page 145

~~F (M) = # spec M dim = i , page 122 i {P ∈ | P }~~

~~F (M) = (F0(M), F1(M),...,Fd(M)), F - vector of M, page 122~~

~~(M) set of all ﬁlters in M, page 25 F (M) = ( (M), ,M,M ×), page 25 F ∩ F ∩ (M) = ( (M),⋆,M ×,M), where F ⋆ G = Filt(F + G), page 26 F ⋆ F Filt(A) ﬁltergeneratedby A, page 26~~

~~F(V ) free binoid on V , page 17~~

FC(V ), FC free commutative binoid on V and 1,...,n , page 17 n { } grA(a) associatedgradedringof A with respect to a ( A, page 108 gr ( ) associated graded binoid of M with respect to ( M, page 108 M I I ht height of , page 122 P P H( ,M) Hilbert-Samuel function of M, page 24 − H(M) Hilbert-Samuel series of M, page 24 hom(M,N) set of all binoid homomorphisms M N, page 14 → 176 Index

~~Hom (A, A′) set of all K - algebra homomorphisms A A′, page 97 K- alg → hom (M,M ′) set of all N - binoid homomorphisms M L, page 49 N →~~

~~I∆ Stanley-Reisner ideal of ∆, page 160 im ϕ image of ϕ, page 14 intc(M) set of all non-integral elements in M, page 19 int(M) set of all integral elements in M, page 19~~

~~/ extended ideal of with respect to M M/ , page 82 J I J → I J(E) = for E spec M, page 116 P∈E P ⊆ KM monoidT algebra of M over K, page 97~~

~~K ideal in KM generated by T a, a , page 99 I ∈ I K[M] binoid algebra of M over K, page 99~~

~~K[ ] ideal in K[M] generated by T a, a , page 99 I ∈ I K[S] K[N] - module of the N -set (S,p), page 105~~

K-Spec A := HomK- alg(A, K) (K -spectrum of A), page 97 ker ϕ kernel of ϕ, page 14 lim Mi direct/inductive limit, page 54 −→ lim Mi projective/inverse limit, page 55 ←− M normalization of M, page 75

~~N M integral closure of M in N, page 75~~

~~M vv bidual of M, page 29~~

~~M v dual of M, page 29~~

~~M × set of all units in M, page 20~~

~~M 0 = M 0 , page 13 ∪{ } M • = M , page 13 \ {∞}~~

~~M∆ binoid associated to ∆, page 160~~

~~M+ set of all nonunits in M, page 20~~

M ∞ = M , page 13 ∪ {∞} M = M F (M) , page 72 F(M) { F | ∈F } M = (M , , ,M), where M M = M , page 72 F(M),◦ F(M) ◦ {∞} F ◦ G F ∩G M (K) algebraof(n n)-matrices, page 102 n × M localization of M at M , P spec M, page 89 P \P ∈ Symbols 177

~~MS localization of M at S, page 70~~

~~Mbool booleanization of M, page 118~~

~~M = M/ for some submonoid S M, page 35 can,S ∼can,S ⊆~~

~~Mcan = Mcan,int M , page 35~~

M = int(M) , page 20 int ∪ {∞} M = M/ , page 34 pos ∼pos M = M/ , page 34 red ∼red M = M/ , page 35 tf ∼tf M/ Rees quotient of M with respect to , page 82 I I map (S,T ) setofallpointedmaps(S,p) (T, q), page 13 p7→q → map S set of all pointed maps (S,p) (S,p), page 13 p → min M set of all minimal prime ideals in M, page 92 min set of all minimal prime ideals of , page 92 M I I mult ( M ,L) set of all N - multi binoid homomorphisms M L, page 51 N i∈I i i∈I i → (Nn)∞ Q = ((Nn) , +, (0,..., 0), ), page 12 Q ∪ {∞} ∞ N = n N n d , page 16 ≥d { ∈ | ≥ } N-spec M set of all N - points of M (N -spectrum of M), page 28

~~N- spec ∆ set of all N -points of ∆ (N -spectrum of ∆), page 155 nil(M) set of all nilpotent elements in M, page 19~~

~~P(V ) powersetof V , page 12~~

~~P(V )∩, P(V )∪ binoids deﬁned by the discrete topology on V , page 12~~

~~P , P binoids deﬁned by the discrete topology on 1,...,n , page 12 n,∩ n,∪ { } Rˆ a - adic completion of the ring R, page 135~~

(1) , i =1, 2, 3, relations on a one-generated binoid, page 57 Ri (2) , i =1, 2, 3, relations on a two-generated binoid, page 60 Ri (s) relation on an s-generated binoid, page 57 R R( ) := (a,b) a b , page 31 ∼ { | ∼ } R[∆] Stanley-Reisner algebra of ∆, page 160 simplB category of simplicial binoids, page 160

~~SC category of simplicial complexes together with simplicial morphisms, page 150~~

~~SCα category of simplicial complexes together with α-simplicial morphisms, page 150 178 Index~~

SCβ category of simplicial complexes together with β-simplicial morphisms, page 150 s - lim Mi strong projective limit, page 56 ←− spec M set of all prime ideals in the binoid M (spectrum of M), page 85 spec∅ M extended spectrum of the binoid M, page 113

~~Spec R set of all prime ideals in the ring R (spectrum of R), page 123~~

~~Spm R set of all maximal ideals in the ring R (maximal spectrum of R), page 125~~

~~(V ) setofallsubset-closedsubsetsofP(V ), page 25 S (V ) , (V ) binoids associated to (V ), page 25 S ∪ S ∩ S supp(f) supportof f, page 18~~

~~Top category of topological spaces, page 117~~

, binoids deﬁned by a topology , page 12 T∩ T∪ T V( ) closed set in the Zariski topology on spec M, page 113 − V ( ) closed set in the Zariski topology on K-spec M, page 123 K − (Zn)∞ = (Zn , +, (0,..., 0), ), n 1, page 20 ∪ {∞} ∞ ≥ (Z/nZ)∞ = (Z/nZ , +, [0], ), n 2, page 20 ∪ {∞} ∞ ≥ Bibliography

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