Projective Quantales and Quantale Modules

Masaryk University Faculty of Science

Master’s Thesis

Radek Slesingerˇ

Advisor: doc. RNDr. Jan Paseka, CSc. Brno, May 2008 Prohl´aˇsen´ı

Prohlaˇsuji,ˇzejsem tuto diplomovou prácivytvoˇrilsamostatnˇea veˇskeré pouˇzitézdroje a literaturu v práciˇrádnˇecituji a uvád´ımv seznamu.

V Brnˇedne 16. kvˇetna2008 Acknowledgement

I would like to thank my advisor Dr. Jan Paseka for directing me during my work on this thesis and providing me with many valuable suggestions and comments that enabled me to complete the thesis. Contents

Introduction 2

1 Preliminaries 3 1.1 Sup-lattices ...... 3 1.2 Quantales ...... 4 1.3 Quantale Modules ...... 5 1.4 Projectivity ...... 6 1.5 Supercontinuous and Superalgebraic Lattices ...... 7

2 Projective Quantales 11 2.1 Free Quantales ...... 11 2.2 Supercontinuous Quantales ...... 12 2.3 Several Elementary Properties of Projective Quantales . . . . 13 2.4 More General View at Projective Quantales ...... 15 2.4.1 K-ﬂat Projective Quantales ...... 15 2.4.2 K-coherent quantales ...... 18 2.4.3 Concrete Instances ...... 19

3 Projective Quantale Modules 21 3.1 Free Modules ...... 21 3.2 Projective Unital Modules ...... 22 3.2.1 Decomposition of Unital Modules ...... 22 3.2.2 Projective Unital Supercontinuous Modules ...... 25 3.3 Relation to Results Obtained for S-posets ...... 27

A Computations of Sup-lattice Endomorphisms 29 A.1 Results ...... 29 A.2 Source Code ...... 31

Bibliography 34

1 Introduction

The notion of quantale, which designates a complete lattice equipped with associative binary multiplication distributing over arbitrary joins, was used for the first time by C. J. Mulvey in 1986. However, multiplicative ordered structures were studied already in 1930s in the form of lattices of ring ideals. During the previous two decades, quantales and quantale modules have found their application in areas of logic, functional analysis, and computer science. Because of a certain analogy between quantales and rings, or quantale and ring modules, respectively, properties like injectivity and projectivity suggest to be studied. Since 1950s a large theory of projective and injective modules over rings has been developed while the problematics of projective rings seems untouched. The case of injective quantales has already been solved in the article [13], showing that the only injective quantale is the trivial one. The authors of the article also attempted to describe projective quantales, however, the proof of the main theorem of the article contained a mistake and even the statement was not valid as can be easily shown. Nevertheless, part of the results can be still used. Also projectivity of sets equipped with an action of a semigroup, both unordered and partially ordered case, has already been studied, and this thesis will attempt to adapt a part of these existing results for some special types of quantale modules. The thesis consists of three chapters. The first chapter introduces basic algebraic and lattice-theoretic notions and general concepts further needed in the following parts. Chapter Two deals with projectivity of quantales. It presents the contemporary knowledge of quantale projectivity and introduces several conditions that projective quantales need to satisfy. The third chapter turns to projectivity of modules over quantales. Following a part of an article that deals with projectivity of certain partially ordered sets, it provides a characteristics of projectivity of a specific class of unital quantale modules.

2 Chapter 1

Preliminaries

This chapter introduces the notions of sup-lattices, quantales and quantale modules (much more on them can be found e.g. in [10] and [4]), the concept of projectivity, and several lattice-theoretic concepts (making use of [3] and [2]) that will be used in both following chapters.

1.1 Sup-lattices

Deﬁnition. A sup-lattice (also complete join-semilattice) is a partially ordered set L where each subset X ⊆ L has its supremum in L.

In this thesis the symbol 1 will denote the top element and 0 will stand for the bottom one. Every sup-lattice is obviously also a complete lattice; the diﬀerence between the categories lies in morphisms.

Deﬁnition. A map f : L → K between two sup-lattices is called a sup- lattice homomorphism if it preserves arbitrary joins, that for any X ⊆ L it satisﬁes _ _ f X = {f(x) | x ∈ X}.

The category of sup-lattices with sup-lattice homomorphisms will be denoted by SLat. When L is a sup-lattice, the set of all sup-lattice endomorphisms L → L with pointwise ordering f ≤ g ⇐⇒ f(x) ≤ g(x) for all x ∈ L and pointwise computed joins forms a sup-lattice again. Let X and Y be posets and f : X → Y , g : Y → X be monotone maps. We say that f is left adjoint to g (and g is right adjoint to f) if f(x) ≤ y ⇐⇒ x ≤ g(y). It is not difficult to check that f preserves all existing suprema, g preserves all existing infima, g is injective iff f is onto, and the adjoint maps are given uniquely. A right adjoint f∗ : K → L can be assigned to every W sup-lattice homomorphism f : L → K as f∗(y) = {x ∈ L | f(x) ≤ y}.

3 1.2 Quantales

Deﬁnition. A quantale Q is a sup-lattice equipped with associative binary multiplication · bound to suprema by the following rules. For any a ∈ Q, X ⊆ Q, it satisﬁes: _ _ a · X = {a · x | x ∈ X}, _ _ X · a = {x · a | x ∈ X}. A quantale Q is called unital if (Q, ·) is a monoid. The multiplicative unit will be denoted by e. Provided that A and B are subsets of a quantale, A · B shall stand for the set {a · b | a ∈ A, b ∈ B}.

Well known examples of quantales include the following:

• ideals of a ring (inclusion ordering and standard multiplication of ideals),

• binary relations on a set (inclusion ordering and composition of relations),

• for a sup-lattice L, Q(L) = {f : L → L | f is a homomorphism} (with pointwise ordering and map composition).

Deﬁnition. A quantale homomorphism is a sup-lattice morphism f : Q → R between two quantales that in addition preserves multiplication. For any X ⊆ Q and x, y ∈ Q: _ _ f X = {f(x) | x ∈ X},

f(x · y) = f(x) · f(y).

If Q and R are unital quantales and f(eQ) = eR, f is called a unital homomorphism.

Within the text Quant will denote the category of quantales with quantale homomorphisms and UnQuant will stand for the category of unital quantales with unital quantale homomorphisms.

Deﬁnition. An element x of a quantale Q is called

• (strictly) right-sided if x·1 ≤ x (x·1 = x). (Strictly) left-sided elements are deﬁned by analogy;

• two-sided if it is both right- and left-sided;

• idempotent if x2 = x.

4 When all elements of a quantale possess some of the properties mentioned above, we use the same name for the quantale. A frame is a quantale in which multiplication is the meet of two elements. It holds that frames are exactly two-sided idempotent quantales: as any elements x, y are two sided, x · y ≤ x and x · y ≤ y, thus x · y ≤ x ∧ y. For the contrary, x ∧ y = (x ∧ y) · (x ∧ y) ≤ (x ∧ y) · y ≤ x · y.

1.3 Quantale Modules

Definition. Let Q be a quantale. A sup-lattice M equipped with a left action ·: Q × M → M is called a left Q-module if the following equalities hold for any q, r ∈ Q, R ⊆ Q, m ∈ M, and L ⊆ M: _ _ R · m = {r · m | r ∈ R}, _ _ q · L = {q · l | l ∈ L}, (q · r) · m = q · (r · m). If Q is unital and e · m = m for any m ∈ M we talk about left unital Q-module. Right modules can be defined in an analogous way. For the rest of the thesis, a module will stand for a left module. Here are several instances of quantale modules: • Any (unital) quantale is a (unital) module over itself. Also the subset of all right-sided elements of a quantale forms a left module and vice versa. • Any sup-lattice L can be turned into a module as well by taking the quantale Q(L) and multiplication f · x = f(x). • As a generalization of the previous example, the set of all maps from a set X to a sup-lattice S forms a Q(S)-module with multiplication f · ϕ = f ◦ ϕ. • If Q is a quantale, Q(Q) can be viewed as Q-module with multiplication (q · f)(x) = q · f(x). Definition. When M and N are Q-modules, a module homomorphism is a sup-lattice homomorphism f : M → N that preserves multiplication by elements of Q. That is, for any q ∈ Q, m ∈ M, X ⊆ M it holds that: _ _ f X = {f(x) | x ∈ X},

f(q · m) = q · f(m). For a given quantale Q, Q-Mod will denote the category of Q-modules with module homomorphisms.

5 1.4 Projectivity

This section shall provide general concepts regarding projectivity that are not speciﬁc for either quantales or modules.

Deﬁnition. An object P of a category C is called projective with respect to a class M of morphisms if for any morphism g : A → B that belongs to M and any morphism f : P → B there exists a morphism h: P → A such that g ◦ h = f as illustrated in the diagram.

P ~ @@ f h ~ @@ ~ @@ ~ @ A g / B In the language of category theory this says that the functor Hom(P, −): C → Set maps M-morphisms to epimorphisms, i.e. surjective maps of sets. A morphism c: B → C is called a regular epimorphism if it is a coequal- izer of some pair of morphisms f and g : A → B. A category is called equationally presentable if all its objects can be deﬁned by a class of operations and equations. It is evident that all the categories Quant, UnQuant and Q-Mod possess this property. As shown in [5], in such a category all regular epimorphisms are exactly surjective morphisms, although epimorphisms need not be surjective in general. This is demonstrated for quantales in [4]. On the other hand, the article [12] shows that regular epimorphisms in Q-Mod are exactly epimorphisms. In this thesis we are going to discuss projectivity of quantales and quantale modules with respect to regular epimorphisms, and so by projectivity we shall mean regular projectivity. One of the sections will then present a view at quantale projectivity with respect to another speciﬁc class of morphisms. Universal properties of free objects and sums yield the following statements.

Proposition 1.1. Let FX be a free object over a set X. Then FX is projective.

Proof. Let ϕ denote the inclusion of X into FX, f : FX → B and g : A → B be homomorphisms, and g be onto. Then picking any g−1(b) ∈ A for each b ∈ (f ◦ ϕ)(X) ⊆ B gives us a map g−1 ◦ f ◦ ϕ: X → A, which induces a homomorphism h: FX → A satisfying g ◦ h ◦ ϕ = f ◦ ϕ. As FX is free over X, ϕ can be cancelled and one obtains the result. L Proposition 1.2. Let P = i∈I Pi. Then P is projective iﬀ each Pi is projective.

6 L Proof. Let Pi, i ∈ I be projectives, κi be their embeddings into i∈I Pi, L f : Pi → B and g : A → B be homomorphisms, and g be onto. As all Pi are projective, there exist homomorphisms hi : Pi → A satisfying L g◦hi = f ◦κi. Universality of the sum yields a homomorphism h: Pi → A such that h ◦ κi = hi for all i. Then we have g ◦ h ◦ κi = g ◦ hi = f ◦ κi, hence g ◦ h = f. Now, let P be projective, κi denote embeddings Pi → P , g : A → B be an onto homomorphism, and fk : Pk → B be a homomorphism for a chosen k ∈ I. For j 6= k put homomorphisms fj : Pj → B arbitrarily (e.g. as constant zero maps, which certainly exist). By sum universality of P there is a homomorphism h: P → B such that h◦κi = fi for all i and projectivity of P gives us a homomorphism l : P → A that satisfies g ◦ l = h. We then have a homomorphism l ◦ κk : Pk → A with g ◦ l ◦ κk = h ◦ κk = fk showing that the chosen Pk is projective. Definition. A morphism f : A → B is called a retraction if there exists a morphism g : B → A such that f ◦ g = idB. The object B is then called a retract of A. The map f is sometimes called a coretraction or section. Using the definition it is easy to see that every retraction is an epimorphism and every coretraction is a monomorphism. Proposition 1.3. Let C be a concrete category where free objects over sets exist and P be an object of C. Then the following conditions are equivalent. 1. P is projective. 2. Every surjective homomorphism with image P is a retraction. 3. P is a retract of a free object. Proof. 1. ⇒ 2. Let f : R → P be a onto and consider the identity map on P . Then there exists a morphism g : P → R, which is a right inversion to f. 2. ⇒ 3. Using the identity on P again together with the map f : P → FP we obtain the unique homomorphism g : FP → P satisfying g ◦ f = idP , which must be onto, and therefore a retraction. 3. ⇒ 1. Let g : S → T be a surjection, f : P → T a morphism, and F a free object with retraction r : F → P and coretraction i: P → F . This situation gives us a morphism h = f ◦ r : F → T and since F is projective, there exists a morphism k : F → S such that g ◦ k = h. The morphism from P to S can be then set as k ◦ i.

1.5 Supercontinuous and Superalgebraic Lattices

Deﬁnition. Let P be a poset. Deﬁne a relation C on P as follows: a C b if for any X ⊆ P that has its supremum in P where b ≤ W X there is

7 some x ∈ X such that a ≤ x. We say that a lies completely below b (in some sources this relation may be also called totally below, way below, or superway below). W From aCb it follows that a ≤ b, considering b ≤ {b}. It also holds that a C b ≤ c or a ≤ b C c implies a C c. Deﬁnition. A complete lattice L is called supercontinuous if each element x of L can be expressed as a join of those elements that are completely below x.

Example. 1. In powersets, to be completely below a set X means to be a one-element subset of X. Powerset lattices are therefore supercontinuous.

2. The real unit interval [0, 1] is supercontinuous with C equal to <. G. N. Raney proved in his article [9] that supercontinuous lattices are exactly completely distributive complete lattices — those complete lattices that satisfy the general distributivity law, i.e. when the identity ^ _ _ ^ xij = xif(i) i∈I j∈J f∈IJ i∈I holds for arbitrary sets I and J. In supercontinuous lattices the relation C is interpolative, that is, a C b implies existence of c satisfying a C c C b [3, p. 56]: consider the set D = S W W W {d | (∃c) d C c C b} = {{d | d C c} | c C b}. Then D = { {d | d C c} | W W c C b} = {c | c C b} = b. As b ≤ D and a C b, there exists d ∈ D greater than or equal to a. An element x is called join-prime if x ≤ a ∨ b implies x ≤ a or x ≤ b. Dualizing propositions I-3.3 up to I-3.13 of [3] one concludes that a supercontinuous lattice is generated by its subset of join-prime elements. Q Proposition 1.4 ([2]). Let L = i∈I Li be a product of complete lattices. Then L is supercontinuous iﬀ each Li is supercontinuous. Proof. Since order on lattice product is given componentwise, for x, y ∈ Q Q i∈I Li we have x C y =⇒ xi C yi for each i, and therefore i∈I Li being supercontinuous implies all Li are supercontinuous. Q Now, if Li are supercontinuous, then i∈I Li is supercontinuous, too, with the relation C given as x C y iﬀ xi C yi for all i and there is at most one Q i such that xi 6= 0. Let x ∈ i∈I Li. For each i there exists yi ∈ Li such that Q yi C xi, so we can set an element zi ∈ i∈I Li as zii = yi and zij = 0 when i 6= j. It is evident that zi C x and x is the supremum of all such elements Q zi, therefore i∈I Li is supercontinuous. This also implies that any element z C x is of the form zi as described above.

8 Proposition 1.5 ([13]). Retracts of supercontinuous lattices are also supercontinuous.

Proof. Let K be a retract of a supercontinuous sup-lattice L realized by a retraction f : L → K and a coretraction g : K → L. First we show that the implication l C g(k) =⇒ f(l) C k (?) holds for any k ∈ K and l ∈ L. If k ≤ W M, then g(k) ≤ g (W M) = W {g(m) | m ∈ M} and l C g(k) implies l ≤ g(m) for some m ∈ M. That means f(l) ≤ (f ◦ g)(m) = m, therefore f(l) C k. The fact that L is supercontinuous says that for every k ∈ K, g(k) = W W {l ∈ L | l C g(k)}. Calculate k = (f ◦ g)(k) = f ( {l ∈ L | l C g(k)}) = W W {f(l) | l ∈ L, l C g(k)} what means k = {m ∈ K | m C k} by (?), and K is therefore supercontinuous, too.

Deﬁnition. An element x of a partially ordered set is said to be supercompact (also hypercompact or completely join-prime) if x C x. A lattice is called superalgebraic if any its element can be expressed as a supremum of supercompact elements. It follows easily that any superalgebraic complete lattice is supercontinuous.

We will denote the set of supercompact elements of a poset P by sc(P ).

Example. 1. Powerset lattices are superalgebraic with supercompact elements represented by singletons.

2. The lattice of ideals of the ring Z/12Z is superalgebraic with supercompact elements (6), (4), and (3).

(1) ? ? (2) (3) LLL LLL (4) (6) ?? (12)

3. The real unit interval [0, 1] is not superalgebraic since the only supercompact element is 0.

As shown in [3], every ﬁnite distributive lattice is generated by its subset of join-prime elements. In case of ﬁnite lattices join-primality means supercompactness, and the lattice is therefore superalgebraic. Q Proposition 1.6 ([2]). Let L = i∈I Li be a product of superalgebraic complete lattices. Then L is superalgebraic too.

9 Proof. By Proposition 1.4, every supercompact element x of L is of the form (xi)i∈I such that xi is supercompact in Li and |{i ∈ I | xi 6= 0}| ≤ 1. As each Li is superalgebraic, any element of the product can be expressed as a join of these supercompact elements. Proposition 1.7. Let L be a superalgebraic complete lattice. Then Q(L), the quantale of all sup-lattice endomorphisms of L, is superalgebraic too.

Proof. For each pair (ci, cj), ci, cj ∈ sc(L), deﬁne a map fij : L → L as ( cj if x ≥ ci, fij(x) = 0 otherwise. W As S is superalgebraic, we can see that fij preserves joins — if X ≥ ci, W W ci ≤ x for some x ∈ X, and therefore x∈X fij(x) = cj = f ( X). The case W of X ci is evident. Each fij is also supercompact in Q(L): Let fij ≤ W W k∈K gk. Namely fij(ci) ≤ k∈K gk(ci). As fij(ci) = cj is supercompact in L, fij(ci) ≤ gk(ci) for some k ∈ K. This inequality then holds for all x ∈ L since gk(x) ≥ gk(ci) ≥ cj = fij(x) if x ≥ ci, and gk(x) ≥ 0 = fij(x) when x ci. Now let f : L → L be an arbitrary homomorphism and x ∈ L. Obviously W f ≥ {fij | fij ≤ f}. The last argument of the previous paragraph also W W yields fij ≤ f iﬀ fij(ci) ≤ f(ci). Then ( {fij | fij ≤ f})(x) = {fij(x) | W W fij ≤ f} = {fij(ck) | fij ≤ f, ck ≤ x, ck ∈ sc(L)} ≥ {fkj(ck) | fkj ≤ W f, ck ≤ x, ck ∈ sc(L)} = {cj | cj ≤ f(x), cj ∈ sc(L)} = f(x). This holds because there exists a corresponding fkj for any ck ≤ x and cj ≤ f(x). By application of the downset functor, which maps every subset A of a poset X to the subset ↓ A = {x ∈ X | (∃a ∈ A) x ≤ a}, any poset (X, ≤) can be turned into a complete lattice (D(X), ⊆). The resulting structure is a superalgebraic lattice with supercompact elements corresponding to principal downsets ↓ x. A monotone map f : X → Y then induces a sup-lattice map D(f): D(X) → D(Y ), which acts as D(f)(A) = ↓ f(A) and preserves S S S S suprema: D(f)( i∈I Ai) = ↓ f i∈I Ai = ↓ i∈I f(Ai) = i∈I ↓ f(Ai) = S i∈I D(f)(Ai). This functor also produces quantales from partially ordered semigroups and unital quantales from partially ordered monoids. With multiplication of downsets U, V ∈ D(S) given as U ∗ V = ↓(U · V ) we can verify that this operation is associative and preserves joins: • (U ∗ V ) ∗ W = ↓(↓(U · V ) · W ) = ↓((U · V ) · W ) = ↓(U · (V · W )) = ↓(U · ↓(V · W )) = U ∗ (V ∗ W ) S S S S • U ∗ i∈I Vi = ↓(U · i∈I Vi) = ↓ i∈I (U · Vi) = i∈I ↓(U · Vi) = S i∈I (U ∗ Vi) Quantales resulting from this construction are supercoherent — their set of supercompact elements is closed under multiplication.

10 Chapter 2

Projective Quantales

Study of projectivity of quantales was initiated by the article [13] where its authors attempted to provide an exact characterization of regular projective quantales and further to develop a larger framework for quantale projectivity. Although they did not succeed to describe regular projective quantales, concepts introduced in that section of the article and most of propositions proved there still can be used. They form the basis of ﬁrst two sections of this chapter. The concepts of quantale projectivity with respect to a certain class of homomorphisms of partially ordered semigroups were further generalized by J. Paseka in [7] and [8], and they are presented in the last section of this chapter.

2.1 Free Quantales

Let X be a set and X+ (X∗) denote the free semigroup (free monoid with the empty word ε, respectively) over X. Then P(X+) with joins given by unions, meets by intersections, and multiplication defined as A · B = {u · v | u ∈ A, v ∈ B} is the free quantale over X. The free unital quantale is then given in the same way with addition of {ε} as the unit of multiplication. + When S is a semigroup and a = a1 . . . an is an element of S , a will designate the product a1 · ... · an ∈ S. Let P be a projective quantale, ϕ denote the inclusion of P into the free quantale P(P +) and consider the identity map on P . Then there exists a + unique quantale homomorphism σ : P(P ) → P , which satisfies σ◦ϕ = idQ. It can be easily verified that σ(A) = W{a | a ∈ A} has the desired property. We already know that projective objects are retracts of free ones. The following proposition shows that we can concentrate at a specific type of retraction.

Proposition 2.1 ([13]). A quantale Q is projective iﬀ there exists a homo- + morphism h: Q → P(Q ) such that σ ◦ h = idQ.

11 Proof. Follows from the fact that σ is onto and from Proposition 1.3.

An analogous statement holds for unital quantales.

2.2 Supercontinuous Quantales

Definition ([13]). Let Q be a supercontinuous quantale. Then Q is called stable if the relation C is stable under multiplication, that is: 1. a C x and b C y imply a · b C x · y, 2. c C x · y implies existence of a, b such that a C x, b C y, and a · b = c. A supercontinuous quantale satisfying the first condition is called weakly stable. In case of weakly stable supercontinuous quantales we have a weakened W W variant of the second condition: c C x · y = {u | u C x}· {v | v C y}, and therefore there exist u C x, v C y with c ≤ u · v. Proposition 2.2 ([13]). The free quantale P(X+) (or the free unital quantale P(X∗)) is supercontinuous and stable. Proof. Since P(X+) is a power set (thus every A ∈ P(X+) can be expressed as a union of singletons that are completely below A), it is supercontinuous. From ∅= 6 A C B it follows that A = {b} for some b ∈ B. Then A C B, C C D, with A, C 6= ∅, give A · C = {b}·{d} = {b · d} C B · D. The case where A = ∅ or C = ∅ obviously holds. Starting with ∅= 6 A C B · C we deduce that A = {b · c} ⊆ B · C, which 0 0 0 0 is equivalent to A = B · C where B C B and C C C. Again, when A = ∅, A = ∅ · ∅ and satisfies the requirement.

Proposition 2.3 ([13]). A retract of a weakly stable supercontinuous quantale is also supercontinuous and weakly stable. Proof. Let f : R → Q and g : Q → R be quantale homomorphisms satisfying gf = idR and Q be supercontinuous and weakly stable. By Proposition 1.5, R is supercontinuous, too, and it remains to show that it is weakly stable as well. Now let x, y, u, v be elements of R such that x C u, y C v. Then both u W W and v can be expressed as u = {g(a) | aCf(u)} and v = {g(b) | bCf(v)} what implies that x ≤ g(a) for some aCf(u), and y ≤ g(b) for some bCf(v). Since x · y ≤ g(a) · g(b) = g(a · b) and a · b C f(u) · f(v) = f(u · v), we can conclude g(a · b) C (g ◦ f)(u · v) = u · v, and thus x · y C u · v. Corollary 2.4. Every projective quantale is supercontinuous and weakly stable.

12 The authors of [13] attempted to show that stability of a supercontinuous quantale implies its projectivity. They deﬁned a map from Q to P(Q+) as + f(x) = {u ∈ Q | uCx in Q}. Unfortunately, this was not a homomorphism as it did not preserve multiplication. Moreover, the text [4] gives an example of a stable supercontinuous quantale, which is not projective. The authors consider the quantale 2 = {0, 1} with 0 < 1 and 1 · 1 = 1, which fails to satisfy necessary conditions listed in the next section.

Theorem 2.5. Every weakly stable supercontinuous frame is projective as unital quantale.

Proof. We shall prove this using a slightly improved version of the map used in [13]. Let Q be a supercontinuous frame, deﬁne f : Q → P(Q∗) as ∗ f(x) = {u ∈ Q | u C x} and verify that it is a homomorphism: W W • Let X ⊆ Q. Then f( X) = {u | u C X}. As the relation C is W interpolative, there exists y ∈ Q such that u C y C X. By deﬁnition of C, y ≤ x for some x ∈ X, thus u C x. Therefore u ∈ f(x) and W S f( X) ⊆ x∈X f(x). W S W • As for all x ∈ X it holds that f(x) ⊆ f( X), x∈X f(x) ⊆ f( X). • All the sets f(x) are monoids because Q is idempotent, hence they contain ε and f(x) ⊆ f(x) · f(y) for any x, y. If u ∈ f(x ∧ y), then u ∈ f(x) ∩ f(y) ⊆ f(x) · f(y).

• Since weak stability implies uv = u ∧ v C x ∧ y for any u ∈ f(x) and v ∈ f(y) we have f(x) · f(y) = {u | u C x}·{v | v C y} ⊆ f(x ∧ y).

This map obviously satisﬁes the property σ ◦ f = idQ since Q is supercontinuous.

2.3 Several Elementary Properties of Projective Quantales

The following propositions list some elementary properties of free (unital) quantales. Since every projective quantale P has a coretraction f going to a free quantale F , we can identify P with f(P ) ⊆ F . Projective quantales then must share certain characteristics of free quantales. As these characteristics are inﬂuenced by unitality, free non-unital and unital quantales need to be distinguished.

Observation. Let A and B be elements of P(X+). 1. If A ⊆ A · B, then A = ∅.

2. The only idempotent in P(X+) is ∅.

13 3. If A · B = ∅, then A = ∅ or B = ∅.

Corollary 2.6. A non-trivial projective quantale is not unital and is inﬁnite.

Proof. Provided that P is projective, it cannot content any other idempotent than 0. Consider the following chain: a = 12 < 1, b = a2 < 1 · 1 = a, c = b2 < a · a = b, etc. If P was ﬁnite, there would exist an element k 6= 0 with k2 = 0. But this is also a forbidden property for a projective quantale.

Observation. Let g : Q → P(Q+) be a quantale homomorphism. Then:

1. Images of elements satisfying x2 ≤ x are subsemigroups of P(Q+). 2. Images of right-sided elements of Q are right ideals of f(1). Images of left- and two-sided elements of possess analogous properties.

Although the quantale 2 mentioned above is not projective in Quant, it is projective as unital quantale. The homomorphisms f : 2 → P({a, b}∗) and g : P({a, b}∗) → 2 are the following: f(0) = ∅, f(1) = {ε}, g(∅) = 0 and g(X) = 1 if X 6= ∅.

Observation. Let A and B be elements of P(X∗).

1. The idempotents in P(X∗) are submonoids of X∗. 2. If A · B = ∅, then A = ∅ or B = ∅.

Proposition 2.7. 1. The only invertible element in a unital projective quantale is e.

2. The top element of a unital projective quantale is idempotent.

Proof. 1. Let P be unital and projective, x · y = e, and f denote an embedding into the free quantale over P . Then f(x) · f(y) = f(e). As f(e) is a monoid, ε ∈ f(e) must have a decomposition into a product of elements of f(x) and f(y), so both f(x) and f(y) contain ε. Thus f(x) ⊆ f(x) · f(x). Since f is an embedding, x ≤ x2. Multiplying this inequality by y and y2 we get e ≤ x and y ≤ e, respectively. But using the same procedure we can obtain the converse inequalities, therefore x = y = e.

2. Obviously 12 ≤ 1. As e ≤ 1 and f(e) contains ε, f(1) contains it too, and f(1) ⊆ f(1) · f(1). Thus 12 = 1.

14 2.4 More General View at Projective Quantales

This section will deal with another form of projectivity introduced by B. Banaschewski in [1] for frames, and further generalized by J. Paseka for quantales [7] and sup-algebras [8]. All the statements in this section come from the article [7] whereas some of the proofs were adapted from [8]. Our basic environment here will be OSgr, the category of ordered semigroups with order-preserving multiplication where morphisms are monotone and preserve multiplication. In this category we will consider a subcategory K that satisfies the following two conditions: 1. Quant is reflective in K. 2. K is corestrictive over Quant. This means that for any K-morphism f : K → Q where K is arbitrary and Q is a quantale, the corestriction of f to a subquantale of Q containing Im(f) also belongs to K. The form of projectivity we are going to discuss here was named as K-flat projectivity by Banaschewski. In our case it designates projectivity with respect to surjective quantale homomorphisms g : Q → R whose right adjoint g∗ : R → Q belongs to K. As K contains Quant reflectively (the reflection functor will be called F ), there exists a universal K-morphism ηA : A → FA for each object A of K. On the other hand, every quantale Q is equipped with a quantale morphism εQ : FQ → Q. These two maps satisfy εQ ◦ ηQ = idQ.

2.4.1 K-ﬂat Projective Quantales

Proposition 2.8. 1. Every FA is generated by the image of ηA.

2. For any quantale Q, ηQ ◦ εQ ≥ idFQ.

3. ηA is order-reﬂecting. That means if f and g are quantale homomorphisms FA → Q, f ◦ ηA ≤ g ◦ ηA implies f ≤ g.

4. If Q is a quantale and h: Q → FQ is a right inverse to εQ, then h ◦ εQ ≤ idFQ.

Proof. 1. Let R be the subquantale of FA generated by Im(ηA), ϕ: A → R be the corresponding corestriction of ηA, and i: R → FA the identical embedding. By corestrictivity and reﬂectivity of Quant in K there exists a quantale homomorphism h: FA → R such that h ◦ ηA = ϕ. Therefore i ◦ h ◦ ηA = i ◦ ϕ = ηA, and universality of ηA yields i ◦ h = idFA. Thus i is onto, and R = FA.

2. By (1) each b ∈ FA is a join of all ηQ(a) ≤ b. Since εQ ◦ ηQ = idQ, ηQ(a) = (ηQ ◦ εQ ◦ ηQ)(a) ≤ (ηQ ◦ εQ)(b) for any such a. Hence W b = ηQ(a) ≤ (ηQ ◦ εQ)(b).

15 3. Let f ◦ηA ≤ g ◦ηA. That means (f ◦ηA)(a) ≤ (g ◦ηA)(a) for all a ∈ A, and consequently f(b) ≤ g(b) for all b ∈ FA.

4. If εQ ◦ h = idQ, by (2) we have h = idFQ ◦ h ≤ ηQ ◦ εQ ◦ h = ηQ, and thus h ◦ εQ ◦ ηQ = h ◦ idQ = h ≤ ηQ = idFQ ◦ ηQ. As both h ◦ εQ and idFQ are quantale homomorphisms, (3) implies the desired inequality.

Remark 2.9. From εQ ◦ηQ = idQ and Proposition 2.8.2 we can deduce that ηQ is right adjoint to εQ. If h: Q → FQ satisﬁes εQ ◦ h = idQ, then h is left adjoint to εQ by Proposition 2.8.4, thus unique. Proposition 2.10. Let g : A → B be a K-morphism and f : Q → R be a quantale homomorphism. Then both diagrams

ηA εQ A / FA and FQ / Q

g F g F f f B / FB FR / R ηB εR commute.

Proof. The fact that Quant is contained in K reflectively makes the first diagram commute. To prove the statement for the second diagram it suffices to show that εR ◦ F f ◦ ηQ = f ◦ εQ ◦ ηQ. Knowing that the left diagram commutes and εQ ◦ ηQ = idQ we can compute f = idR ◦ f = εR ◦ ηR ◦ f = εR ◦ F f ◦ ηQ and f = f ◦ idQ = f ◦ εQ ◦ ηQ. Using Proposition 2.8.3 we obtain the desired result.

Proposition 2.11. The reﬂection functor F preserves partial order of maps, and F ηA is left adjoint to εFA in Quant. Proof. Let f, g : A → B be K-morphisms, which satisfy f ≤ g. Using the previous proposition we can see that F f ◦ ηA = ηB ◦ f ≤ ηB ◦ g = F g ◦ ηA, and therefore by Proposition 2.8.3 again, F f ≤ F g. Now consider the following diagram:

ηA A / FA

ηA F ηA FA / FFA / FA ηFA εFA

It follows that idFA ◦ ηA = εFA ◦ ηFA ◦ ηA = εFA ◦ F ηA ◦ ηA. Thus idFA = εFA ◦ F ηA. The assertion is then implied by Remark 2.9.

We can deﬁne a relation on each quantale Q as y CQ x iﬀ x ≤ εQ(z) =⇒ ηQ(y) ≤ z for all z ∈ FQ.

16 Proposition 2.12. Let Q and R be quantales, x, y, u, v ∈ Q, and f : Q → R be a quantale isomorphism. Then

1. x ≤ y CQ u ≤ v implies x CQ v and x ≤ v,

2. f(y) CR f(u) iﬀ y CQ u.

Proof. 1. Suppose v ≤ εQ(z) for some z ∈ FQ. Then u ≤ v ≤ εQ(z) implies x ≤ y ≤ ηQ(z).

2. Let f(u) ≤ εR(z) for some z ∈ FQ and consider the following diagram:

εQ ηQ FQ / Q / FQ

F f f F f FR / R / FR εR ηR

−1 As F f and F f are isomorphisms of quantales, εR(z) = (f ◦ εQ ◦ −1 −1 F f )(z), hence u ≤ (εQ ◦ F f )(z). Whence it follows that ηQ(u) ≤ −1 F f (z), and so (ηR ◦ f)(u) = (F f ◦ ηQ)(u) ≤ b. Since f is an isomorphism, the converse implication proceeds in the same way.

Theorem 2.13. The following conditions are equivalent for any quantale Q:

1. Q is K-ﬂat projective.

2. εQ has a right inverse. 3. Q is a retract of FA for some A from K. W 4. Every x ∈ Q satisﬁes x = {y ∈ Q | y CQ x}, and x CQ a, y CQ b imply x · y CQ a · b.

Proof. 1. ⇒ 2. By Remark 2.9, εQ has the right adjoint ηQ, and it is onto since εQ ◦ ηQ = idQ. Therefore, if Q is projective, there exists a quantale homomorphism h: Q → FQ such that εQ ◦ h = idQ. 2. ⇒ 3. This is evident. 3. ⇒ 1. As projectivity is preserved by retraction, we only need to show that each FA is K-ﬂat projective. Consider the following diagram where f and k are quantale homomorphisms, k is onto and has a right adjoint in K.

ηA A / FA | k fη | f ∗ A | g }| Q / / R k

17 Since k∗ ◦ f ◦ ηA is in K, there exists a quantale homomorphism g : FA → Q that satisfies g◦ηA = k∗ ◦f ◦ηA. Applying k we have k◦g◦ηA = k◦k∗ ◦f ◦ηA. The right hand side equals f ◦ ηA because k is onto, thus k ◦ k∗ = idR. Proposition 2.8.3 then yields k ◦ g = f. 2. ⇒ 4. Let hQ : Q → FQ satisfy εQ ◦ hQ = idQ. We will show that y CQ x iff ηQ(y) ≤ hQ(x) for the given hQ. Since x = (εQ ◦ hQ)(x), setting z = hQ(x) and looking at the definition of CQ we obtain ηQ(x) ≤ hQ(x). To show the converse implication, we will use Proposition 2.8.4. From x ≤ εQ(z) it follows that hQ(x) ≤ (hQ ◦ εQ)(z) ≤ z. Thus, ηQ(y) ≤ hQ(x) implies ηQ(y) ≤ z. W By Proposition 2.8.1, hQ(x) = {ηQ(y) | ηQ(y) ≤ hQ(x)}. This yields W W x = (εQ ◦ hQ)(x) = εQ ( {ηQ(y) | ηQ(y) ≤ hQ(x)}) = {(εQ ◦ ηQ)(y) | y CQ x}. Now, let xCQ a and yCQ b. Using the previous equivalence we can derive ηQ(x·y) = ηQ(x)·ηQ(y) ≤ hQ(a)·hQ(b) = hQ(a·b) what means x·y CQ a·b. W 4. ⇒ 2. Define a map hQ : Q → FQ as hQ(x) = {ηQ(y) | y CQ x}, W W which yields (εQ◦hQ)(x) = {(εQ◦ηQ)(y) | yCQx} = {y | yCQx} = x. As W yCQεQ(x) implies ηQ(y) ≤ x, we get (hQ◦εQ)(x) = {ηQ(y) | yCQεQ(x)} ≤ x, and therefore hQ ◦ εQ ≤ idQ. This says that hQ is left adjoint to εQ, and is join-preserving. Multiplication is preserved as well since we have these inequalities: W W W • hQ(x) · hQ(y) = {ηQ(u) | u CQ x}· {ηQ(v) | v CQ y} = {ηQ(u) · W W ηQ(v) | u CQ x, v CQ y} = {ηQ(u · v) | u CQ x, v CQ y} ≤ {ηQ(w) | w CQ x · y} = hQ(x · y)

• hQ(x · y) = hQ((εQ ◦ hQ)(x) · (εQ ◦ hQ)(y)) = hQ(εQ(hQ(x) · hQ(y))) ≤ hQ(x) · hQ(y)

2.4.2 K-coherent quantales

An element x ∈ Q is called K-compact if x CQ x. Let cK(Q) denote the set of all K-compact elements of a quantale Q. Then Q is called K-coherent if it satisﬁes the following conditions: W 1. for all x ∈ Q, x = {c ∈ cK(Q) | c ≤ x},

2. if c1, c2 ∈ cK(Q), then also c1 · c2 ∈ cK(Q). Proposition 2.14. Let K have equalizers coinciding with equalizers in Set and let Q be K-ﬂat projective. Then cK(Q) belongs to K.

Proof. By Theorem 2.13, proof of 2. ⇒ 4., cK(Q) = equal(ηQ, hQ). The condition above is satisﬁed e.g. when K is equationally presentable.

18 Proposition 2.15. The quantale FA is K-coherent for any A ∈ K, and the set cK(FA) contains the set ηA(A).

Proof. By Proposition 2.10, F ηA is left adjoint to εFA in Quant, and (F ηA ◦ ηA)(a) = (ηFA ◦ ηA)(a) for any a ∈ A. Then hFA = F ηA, hence (hFA ◦ ηA)(a) = (ηFA ◦ ηA)(a), and the inclusion is shown. We also know that FA is K-ﬂat projective and as such it is join-generated by ηA(A), and therefore it is K-coherent.

2.4.3 Concrete Instances The authors of [1], [7] and [8] used the following way to specify subcategories of OSgr (meet-semilattices or partially ordered algebras, respectively): for each po-semigroup A, let SA denote the collection of subsets of A that satisfy: {a · s | s ∈ S} and {s · a | s ∈ S} belong to SA for each a ∈ A and S ∈ SA, and f(S) ∈ SB for each po-semigroup homomorphism f : A → B and S ∈ SA. Now, let S denote the subcategory of OSgr that contains all semigroups A such that W S exists for each S ∈ SA, a · W S = W{a · s | s ∈ S}, W S · a = W{s · a | s ∈ S}, and all po-semigroup homomorphisms preserve all W S. This construction can be illustrated in several examples, depending on what SA contains for each A:

1. When SA is empty, S = OSgr. This setting was studied in the second part of the article [13], and the authors obtained speciﬁc versions of statements presented in [7] and subsequently here.

2. When SA contains all ﬁnite subsets of A, S is the category of m- semilattices (po-semigroups closed under all ﬁnite joins and multiplication distributing over the joins).

3. When SA contains all subsets of A, S = Quant.

It is evident that all these categories contain Quant, and they are corestrictive over quantales.

Proposition 2.16. Quant is reﬂective in any S.

Proof. For any A ∈ S, take the downset quantale D(A). Let γA be a system of all such elements U of D(A) that S ⊆ U and S ∈ SA imply W S ∈ U. As γA contains all principal downsets ↓ a, we can set a map σA : A → γA as σA(a) = ↓ a. For any A ∈ S it then holds that: 1. γA is a quantale,

2. σA is the universal map in S to quantales.

19 Proofs of these statements would follow their analogies in [1] and [7].

In the examples shown above the reﬂections γA consist of following sorts of downsets of A:

1. all downsets,

2. ideals of A,

3. principal downsets ↓ a.

For any quantale Q, the adjunction εQ : γQ → Q is the join map: W εQ(U) = U. Using these characterizations we can derive the concrete W form of the relation CQ: y CQ x iﬀ x ≤ U implies ↓ y ⊆ U for any downset U ∈ γQ. Concerning S = OSgr, this is exactly our well-known relation “completely below”; on the other hand, when S = Quant, we have CQ equal to ≤. ∼ Proposition 2.17. γA is S-coherent for any A ∈ S, and A = σA(A) = cS (γA).

Proof. Proposition 2.15 implies the first part and the inclusion σA(A) ⊆ cS (γA). S Now, let U ∈ γA. Then u∈U σγA(↓ u) ∈ γ(γA). Let {Ij | j ∈ J} be a collection of downsets in γA that belongs to S(γA) and {Ij | j ∈ S J} ⊆ u∈U σγA(↓ u). For any j ∈ J then Ij ∈ σγA(↓ uj) for some uj ∈ U, W and therefore Ij ⊆ ↓ uj. Put vj = Ij ≤ uj. As εγA : γ(γA) → γA is a po-semigroup homomorphism, the definition of SA yields εγA({Ii | j ∈ W J}) = {vj | j ∈ J} ∈ SA. Denoting {vj | j ∈ J} by v, we have v ∈ U because of the definition of γA, thus Ij ⊆ ↓ v and Ij ∈ σγA(↓ v) for any j ∈ J. From this it follows that {Ij | j ∈ J} ⊆ σγA(↓ v). Therefore W S {Ij | j ∈ J} ∈ σγA(↓ v) ⊆ u∈U σγA(↓ u). S S Let U CγA U = εγA u∈U σγA(↓ u) . Then σγA(U) ⊆ u∈U σγA(↓ u). S Thus U ∈ u∈U σγA(↓ u), and therefore U ∈ σγA(↓ u) for some u ∈ U. This implies U ⊆ ↓ u ⊆ U, concluding cS (γA) ⊆ σA(A). Unfortunately, these results are not applicable to the case of regular projectivity — a surjective quantale homomorphism has a right adjoint, however, this adjoint does not preserve multiplication in general. It suffices to consider a constant homomorphism from a 2-element quantale {0, 1} with trivial multiplication 1 · 1 = 0 to the trivial quantale. Then W{0, 1} = 1 6= 1 · 1. As we can also see, analogous results can be achieved when considering unital quantales and K as a subcategory of the category of partially ordered monoids.

20 Chapter 3

Projective Quantale Modules

In the main section of this chapter we follow the part of the article [11] where projectivity was studied in the category of so called S-posets, partially ordered sets equipped with action of a partially ordered monoid compatible with order relation. In accordance with the article we ﬁrst develop a little theory of decomposability, and then we apply it to obtain the main result.

3.1 Free Modules

Quantale modules were studied in the thesis [6] (in Czech) where detailed proofs of the following propositions can be found. L Proposition 3.1. The sum i∈I Mi is isomorphic to the product of Mi. In fact, such a proposition holds even for sup-lattices in general.

Proposition 3.2. If Q is a unital quantale, the free Q-module over {x} is isomorphic to Q considered as a module.

Consider the the set 2×Q where 2 stands for {0, 1} with 0 < 1. Equipped with multiplication given as ( 0 if h = 0, r · (h, q) = (0, (r · q) ∨ z(r, h)) where z(r, h) = r if h = 1,

2 × Q becomes a Q-module.

Proposition 3.3. The module 2×Q is the free Q-module over a one-element set.

As the free functor preserves colimits, and colimits in Set are unions, the Q free unital Q-module over a set X is isomorphic to x∈X Q, and the free Q Q-module over X is isomorphic to x∈X (2 × Q).

21 3.2 Projective Unital Modules

3.2.1 Decomposition of Unital Modules Let Q be a unital quantale and M be a left unital module over Q. In order to be able to derive results analogous to those presented in [11] we need to impose an additional assumption on M — to be supercontinuous.

Deﬁnition. A submodule N of a module M is called C-closed if a ∈ N and b C a imply b ∈ N. A module M is called decomposable if there exist two nontrivial C-closed submodules of M, A and B, A ∩ B = {0} such that they generate M as sup-lattice. In other terms, M = hA ∪ Bi. If this does not happen, we say that M is indecomposable. Proposition 3.4. Let M be a unital Q-module and m ∈ M be a join-prime element. Then the submodule Qm = {q · m | q ∈ Q} is indecomposable. Proof. Suppose that Qm is decomposable, that is, there exist A and B, submodules of Qm, A ∩ B = 0 such that m = a ∨ b for some a ∈ A, b ∈ B. As m is join-prime, we have m = a or m = b. Without loss of generality we can suppose m = a, thus m ∈ A, b = 0, and Qm ⊆ A.

If M is supercontinuous, the set ↓ Qm = {n ∈ M | n ≤ q·m for some q ∈ Q} with m being join-prime is C-closed (obviously) and indecomposable, too. Assuming ↓ Qm is decomposable into A and B, the same argument as in the previous proposition yields Qm ⊆ A. As A is C-closed, it contains all join-prime elements yj lying completely below any y ∈ A. Let x ≤ y for some y ∈ Qm. Since M is supercontinuous, x is a join of a collection of W join-prime elements xi with all xi C x. Each xi then satisﬁes xi C y = yj, W and thus xi ≤ yj for some j. It then follows that xi = x ∈ A. S Lemma 3.5. Let a module M have decompositions M = h i∈I Aii = S h j∈J Bji. Then for each i ∈ I, the submodule Ai has a decomposition S Ai = h j∈J (Ai ∩ Bj)i. S Proof. It is evident that Ai ⊇ h j∈J (Ai ∩ Bj)i. The converse inclusion also holds: if a ∈ Ai, it is a join of elements bk ∈ Ai where each bk is contained S in some Bj since j∈J Bj generates the whole M.

Proposition 3.6. Let Mi, i ∈ I, be a family of C-closed indecomposable T submodules of a supercontinuous module M satisfying i∈I Mi 6= {0}. Then S the submodule join-generated by i∈I Mi is also C-closed and indecomposable. S S Proof. First, i∈I Mi is C-closed. If m ∈ h i∈I Mii, it is equal to a supre- S mum of a subset N of i∈I Mi where each nj ∈ N is a join of join-prime elements pij ∈ Mi. When nCm, interpolativity of C yields l such that nClCm. S Then l ≤ pij for some i and j, hence n C pij and n ∈ Mi ⊆ h i∈I Mii.

22 S Suppose h Mii is decomposable into A and B. There certainly exists T (because Mi are C-closed) a join-prime element m ∈ Mi, diﬀerent from 0. If it was made as a ∨ b for some a ∈ A and b ∈ B, it would equal either a or b. Again, without losing generality, if m ∈ A, Mi ∩ A 6= {0} for any i. By Lemma 3.5, Mi = h(Mi ∩ A) ∪ (Mi ∩ B)i. As all Mi are indecomposable, S Mi ∩ B = {0}, Mi = hMi ∩ Ai ⊆ A, therefore h Mii = A. Theorem 3.7. Every supercontinuous unital Q-module can be uniquely decomposed into a collection of its Q-submodules, which are C-closed, indecomposable, and pairwise meeting in 0 only.

Proof. We already know that ↓ Qm is C-closed and indecomposable if m is join-prime. Thus, for any join-prime element x, the set Dx = {N | x ∈ N,N is -closed and indecomposable} is nonempty, and T N 6= {0}. C N∈Dx From Proposition 3.6 it follows that the set A = hS Ni is -closed x N∈Dx C and indecomposable, too. Considering x 6= y, either Ax ∩ Ay = {0}, or Ax = Ay. This holds because if there exists m ∈ Ax ∩ Ay, m 6= 0, also a join-prime element n C m belongs to the intersection. Obviously Ax ⊆ hAx ∪ Ayi. Using Proposition 3.6 again, hAx ∪ Ayi is C-closed and indecomposable (since Ax ∩ Ay 6= {0}) containing y, so hAx ∪Ayi ⊆ Ay (as Ay includes all indecomposable C-closed submodules containing y). The converse inclusion can be shown in the same way. We can therefore set an equivalence θ on join-prime elements of M as xθy iﬀ Ax = Ay. Since every element of M is a supremum of join-primes, S M = h x∈C Axi where C is a suitable set of θ-classes representants. S Suppose there exist two such decompositions, namely M = h i∈I Aii = S S h j∈J Bji, and consider one of the Bj. By Lemma 3.5, Bj = h i∈I (Ai∩Bj)i. Let b ∈ Bj be nonzero. It equals to the supremum of join-primes ak, ak C b for all k. Pick some ak of them and the submodule Aik where ak belongs. We then obtain a decomposition of B as h(A ∩B )∪hS (A ∩B )ii. As B j ik j ak∈/Al l j j is indecomposable and A ∩ B is nontrivial, hS (A ∩ B )i ∩ B = {0}, ik j ak∈/Al l j j

Bj = hAik ∩ Bji, and thus Bj ⊆ Aik . And vice versa, inclusion of Aik in some Bm (which is necessarily the considered Bj) can be shown. Identity of the collections Ai, i ∈ I, and Bj, j ∈ J, follows. Corollary 3.8. Every supercontinuous unital quantale has a unique decomposition into a sum of its sub-sup-lattices that are downward closed and closed under left multiplication by elements of the quantale.

Example. Consider the lattice of ideals of the ring Z60, which looks as follows.

23 (1) mm PPP mmm PP mmm PPP (3)mm (2) P(5) @@ nn NNN jjj @@ nnn jNjNjj @nnn jjjj NNN nnn jjj N (6)N (15) (4)L (10) NN =pp LL r NN pp= LL rr pNpN == rrLrL ppp NN rr L (12) (30) (20) PPP p PP ppp PPP pp P(0)pp The join-prime elements of L = Idl(Z60) are a = (12), b = (15), c = (20), and d = (30). A module structure can be introduced on L in several ways. Since it is also a unital quantale, module multiplication can be given naturally, and the cyclic submodules then look like La = {(0), (12)}, Lb = {(0), (15), (30)}, Lc = {(0), (20)}, Ld = {(0), (30)}. As all of them are subchains connecting 0 and the respective elements, they are identical to their downsets. The resulting decomposition is then Aa = La, Ab = Ad = Lb, Ac = Lc. A diﬀerent-looking module results from the construction of the endomorphism quantale Q(L). Multiplication by a quantale element is then performed by homomorphism application. All elements of L can be achieved from any join-prime element, hence all cyclic submodules Q(L)m generated by join-prime elements are equal to L, and L is therefore indecomposable as Q(L)-module by Proposition 3.4. Now look at the quantale Q(L). As L is superalgebraic, Proposition 1.7 says that Q(L) is superalgebraic as well. Since L is join-generated by its set of supercompact elements, any endomorphism f ∈ Q(L) can be prescribed by setting images of those elements. This attribution can be almost arbitrary, it must only preserve the partial order of sc(L). The sup-lattice Q(L) is then isomorphic to the sup-lattice C = {g : sc(L) → L | g is monotone}. As the poset sc(S) can be viewed as a disjoint union of “maximal con- nected components” Si (with respect to the ordering relation), and sums in Pos are exactly disjoint unions, we can derive the following decomposition: ∼ S˙ ∼ Q(L) = HomSLat (L, L) = HomPos (sc(L),L) = HomPos i∈I Si,L = ∼ Q L = i∈I HomPos (Si,L) = i∈I HomPos (Si,L). Besides being a sup-lattice, each Ci = HomPos (Si,L) is also a module because any composition f ◦ g of f ∈ Q(L) and g ∈ Ci is a monotone map Si → L again. Applying this result to the module we were dealing with we can see that 2 2 Q(Idl(Z60)) can be decomposed as Idl(Z60) ⊕ Idl(Z60) ⊕ Idl(Z60) where S means the poset of all monotone maps from 2 to S. This part of the example originates in an observation of the results the author obtained when trying to decompose a module using the procedure

24 suggested by previous propositions. A computer program used for the cal- culations and example results can be found in Appendix A.

3.2.2 Projective Unital Supercontinuous Modules Lemma 3.9. Let Q be a quantale and d ∈ Q be an idempotent. Then the module Qd is projective.

Proof. Let f : Qd → N be a homomorphism and g : M → N be a surjective homomorphism. If f(d) = n ∈ N, there exists m ∈ M such that g(m) = n. Deﬁne h: Qd → M as h(q·d) = q·d·m. This map is a module homomorphism W because h(r · (q · d)) = r · (q · d) · m = r · h(q · d) and h i∈I (qi · d) = W W W i∈I (qi · d) · m = i∈I (qi · d · m) = i∈I (h(qi · d)). The fact of d being idempotent makes the homomorphisms commute: (g◦h)(q·d) = g(q·d·m) = q · d · g(m) = q · d · n = q · d · f(d) = f(q · d · d) = f(q · d).

Especially, considered as modules, unital quantales are projective. From now we will deal only with unital modules. The following lemma provides characterization of projective cyclic unital modules.

Proposition 3.10. Let M be a unital Q-module and m ∈ M. Then the following conditions are equivalent:

1. Qm is projective.

2. There exists an idempotent d ∈ Q such that m = d·m and q ·m 7→ q ·d is a homomorphism.

3. Qm =∼ Qd for some idempotent d ∈ Q.

Proof. 1. ⇒ 2. Let Qm be projective. Since ψ : Q → Qm taking q to q ·m is onto, by Proposition 1.3 it is a retraction and there exists g : Qm → Q such that ψ◦g = idQm. Let d ∈ Q denote the g-image of m. Then m = ψ(g(m)) = ψ(d) = d · m, and d is idempotent: d = g(m) = g(d · m) = d · g(m) = d2. It can be easily verified that the map ϕ: Qm → Q defined as ϕ(q · m) = q · d is a module homomorphism. 2. ⇒ 3. The image of the homomorphism ϕ defined in the previous step is Qd, and ϕ is injective: if q · m, r · m ∈ Qm are such elements that q ·d = r ·d, then q ·m = q ·(d·m) = (q ·d)·m = (r ·d)·m = r ·(d·m) = r ·m. 3. ⇒ 1. See the previous proposition.

A module M is called simple if it is nontrivial, and its only submodules and quotients are M itself and the trivial module. The previous results yield the following:

Corollary 3.11. A simple unital Q-module is projective iﬀ it is isomorphic to Qd for some idempotent d ∈ Q.

25 Proposition 3.12. A supercontinuous, indecomposable unital Q-module is projective if and only if it is isomorphic to Qe for some idempotent e ∈ Q.

Proof. Necessity is implied by Proposition 3.9. Suppose then a Q-module P is projective and indecomposable. Therefore, P is a retract of the free Q Q module i∈I Q with the retraction f : i∈I Q → P and the coretraction Q Q g : P → i∈I Q. This gives us a submodule g(P ) ⊆ i∈I Q isomorphic to Q Q P . More specifically, g(P ) = ( i∈I Q) ∩ g(P ) = i∈I (Q ∩ gi(P )). Suppose there is an element 0 6= x ∈ g(P ) with gi(x) and gj(x) different from 0 for distinct i, j ∈ I. This would give us a decomposition of g(P ) being join-generated by R = {x ∈ g(P ) | xj = 0 for j 6= i} and S = {x ∈ g(P ) | xi = 0} (both R and S are nontrivial by the assumption). But as g(P ) is indecomposable, such a splitting is not possible, and g(P ) is contained in a copy of Q for some i ∈ I. To finish the proof, P = (f ◦ g)(P ) ⊆ f(Qi) ⊆ P , and thus P = f(Q) = Q · f(e) = Qa for f(e) = a ∈ P . By part 3. of Proposition 3.10, P =∼ Qd for an idempotent d ∈ Q.

Theorem 3.13. A supercontinuous unital Q-module is projective if and only Q if it is isomorphic to i∈I Qdi where each di is an idempotent element of Q.

Proof. Necessity follows from the previous proposition and from the fact that products of Q-modules coincide with their sums. It is evident that supercontinuity is not a necessary assumption in this case. For the converse, we have seen that every supercontinuous Q-module M has a unique decomposition into a set of its indecomposable submodules Mi, i ∈ I, making M isomorphic to their sum. As a sum of objects is projective iﬀ every single one is projective, each Mi has to be isomorphic to Qdi for an idempotent di ∈ Q. The example of decomposition of a quantale of sup-lattice endomorphisms at page 23 illustrates this result. All summands can be expressed as cyclic submodules of the quantale, whereas their generators need not be necessarily unique. Details can be found in Appendix A. Since products of supercontinuous lattices are supercontinuous, too, and supercontinuity is preserved by retraction, the previous theorem holds especially for all unital modules over supercontinuous unital quantales.

Corollary 3.14. Let Q be a supercontinuous unital quantale with the only nonzero idempotent element d. Then every projective unital module over Q is free.

Proof. Any projective Q-module has a decomposition as shown above consisting only of copies of Q since there is no other idempotent in Q diﬀerent from 0 than the unit.

26 3.3 Relation to Results Obtained for S-posets

The results obtained in previous two sections provide a certain extension of the ones presented in [11] for the case of S-posets. A partially ordered monoid (shortly po-monoid) S is a monoid with the unit e equipped with a partial order relation compatible with multiplication, that is, for any s, t, u ∈ S, s ≤ t implies s · u ≤ t · u and u · s ≤ u · t. A (left) S-poset is then a partially ordered set X with a left order- compatible action of a po-monoid S that for any x, y ∈ X and s, t ∈ S satisfies: x ≤ y =⇒ s · x ≤ s · y, s ≤ t =⇒ s · x ≤ t · x, (s · t) · x = s · (t · x), e · x = x. An S-poset homomorphism f : X → Y is a monotone map that preserves multiplication by elements of S, i.e. f(s · x) = s · f(x) for any s ∈ S and x ∈ X. By application of the downset functor on both a S-poset X and the po-monoid S we obtain a D(S)-module D(X) with multiplication defined as U ∗ A = ↓(U · A) for U ⊆ D(S) and A ⊆ D(X) (its required properties can be verified in the same way as for downset quantales). An S- poset homomorphism f : X → Y then maps to a sup-lattice homomorphism D(f): D(X) → D(Y ), which preserves the action of D(S): D(f)(U ∗ A) = ↓ f(U ∗ A) = ↓ f(↓(U · A)) = ↓(U · ↓ f(A)) = ↓(U ·D(f)(A)) = U ∗ D(f)(A). Suppose we have a D(S)-module M. Then we can make M be an S-poset by putting s · m = (↓ s) ∗ m while retaining the original ordering.

Proposition 3.15. Let X be a projective S-poset. Then the D(S)-module D(X) is also projective.

Proof. Let M and N be D(S) modules, f : M → N and g : D(X) → N be module homomorphisms, and f be onto. Both f and g induce monotone maps f : M → N and g : sc(D(X)) → N where f = f and g = g|sc(D(X)). Since M and N can be viewed as S-posets, f and g become also S-poset homomorphisms: we have f(s · m) = f((↓ s) ∗ m) = f((↓ s) ∗ m) = (↓ s) ∗ f(m) = s · f(m) = s · f(m), and g(s · (↓ x)) = g((↓ s) ∗ (↓ x)) = g((↓ s) ∗ (↓ x)) = (↓ s) ∗ g(↓ x) = (↓ s) ∗ g(↓ x) = s · g(↓ x) (here we use the fact that ↓ s ∗ ↓ x = ↓(↓ s · ↓ x) = ↓(s · x), thus application of g makes sense). Since X is projective, there is a S-poset homomorphism h: sc(D(X)) → M, which satisﬁes f ◦ h = g. As D(X) is superalgebraic, every A ∈ D(X) S can be expressed as A = a∈A ↓ a, so extend h to h: D(X) → M as h(A) = W a∈A h(↓ a). Indeed, the deﬁnition of h does not depend on the choice S S of generating supercompact elements: let A = i∈I ↓ ai = j∈J ↓ bj. As

27 S each ↓ ai is supercompact, and ↓ ai ⊆ j∈J ↓ bj, there exists k(i) ∈ J for every i such that ↓ ai ⊆ ↓ bk(i). Hence, g(↓ ai) ≤ g(↓ bk(i)), and therefore W W W i∈I g(↓ ai) ≤ i∈I g(↓ bk(i)) ≤ j∈J g(↓ bj). The converse inequality can be obtained in the same way, thus h is well-deﬁned. We can see that h is a D(S)-module homomorphism: • h S A = h S ↓ a = W h(↓ a) = W h(A ) i∈I i i∈I, a∈Ai i∈I, a∈Ai i∈I i • Let x ∈ U ∗ A. Then x ≤ u · a for some u ∈ U, a ∈ A, and h(↓ x) ≤ h(↓(u · a)) = h(↓(u · a)) = u · h(↓ a) = u · h(↓ a) ≤ U ∗ h(A). W W • Let x ≤ U ∗ h(A). Then x ≤ u · a∈A h(↓ a) = a∈A(u · h(↓ a)) = W W a∈A h(u · ↓ a) = a∈A h(↓(u · a)) ≤ h(U ∗ A). W W And ﬁnally, f ◦ h = g because (f ◦ h)(A) = f a∈A h(↓ a) = a∈A(f ◦ W W W S h)(↓ a) = a∈A(f ◦ h)(↓ a) = a∈A g(↓ a) = a∈A g(↓ a) = g a∈A ↓ a = g(A).

28 Appendix A

Computations of Sup-lattice Endomorphisms

In order to determine the decomposition of the unital quantale Q(Idl(Z12)) in the last part of the example in Section 3.2.1, a collection of functions written in Haskell programming language was used. It was created ad-hoc, without extensibility and efficiency in mind, and benefits from the fact that comparison of elements and computing their finite joins can be performed simply using integer operations. The computations were performed using the following program compiled by GHC Haskell compiler version 6.8.2. Function definitions should com- ply to Haskell 98 language specification, and require no additional libraries. When compiled and run, the program writes the results to the standard output. However, the output is quite close to the internal data representation and requires additional processing. Another variant is calling individual functions in an interpreter. In the program each ideal of Zn is represented by the respective divisor of n with an exception of Zn denoted by 0. Sup-lattice homomorphisms are represented as collections of pairs of the form (ideal,image) using a data type provided by the standard library.

A.1 Results

This section presents results obtained by the program for Q(Idl(Z12)). To maintain space eﬃciency, a homomorphism is written here as a six-tuple with the ﬁxed order: [f(0), f(6), f(4), f(3), f(2), f(1)]. By calling joinPrimes endIdlZ 12 one can obtain the following join- prime elements of Idl(Z12): [0, 0, 3, 0, 3, 3], [0, 0, 4, 0, 4, 4], [0, 0, 6, 0, 6, 6], [0, 3, 0, 3, 3, 3], [0, 4, 0, 4, 4, 4], [0, 6, 0, 6, 6, 6], [0, 0, 0, 3, 0, 3], [0, 0, 0, 4, 0, 4], [0, 0, 0, 6, 0, 6], [0, 0, 0, 0, 0, 0].

29 Downsets of join-prime generated cyclic submodules can be obtained by calling the function map ((ds endIdlZ 12).(cyclic endIdlZ 12))(joinPrimes endIdlZ 12) It takes the set of join-primes, for each its element m computes the cyclic submodule Q(Idl(Z12))m, and then the downset of the submodule. The result is a 10-element set containing the following sets of maps. With exception of the one-element set containing the constant zero map, each of four distinct sets appears three times in the output. Apparently, submodules in 1–3. (call them A) and 4–6. (B) meet only in the constant zero map while 7–9. (C) are submodules of B.

1–3. {[0, 0, 1, 0, 1, 1], [0, 0, 2, 0, 2, 2], [0, 0, 3, 0, 3, 3], [0, 0, 4, 0, 4, 4], [0, 0, 6, 0, 6, 6], [0, 0, 0, 0, 0, 0]}

4–6. {[0, 1, 0, 1, 1, 1], [0, 2, 0, 1, 2, 1], [0, 2, 0, 2, 2, 2], [0, 3, 0, 1, 3, 1], [0, 3, 0, 3, 3, 3], [0, 4, 0, 1, 4, 1], [0, 4, 0, 2, 4, 2], [0, 4, 0, 4, 4, 4], [0, 6, 0, 1, 6, 1], [0, 6, 0, 2, 6, 2], [0, 6, 0, 3, 6, 3], [0, 6, 0, 6, 6, 6], [0, 0, 0, 1, 0, 1], [0, 0, 0, 2, 0, 2], [0, 0, 0, 3, 0, 3], [0, 0, 0, 4, 0, 4], [0, 0, 0, 6, 0, 6], [0, 0, 0, 0, 0, 0]}

7–9. {[0, 0, 0, 1, 0, 1], [0, 0, 0, 2, 0, 2], [0, 0, 0, 3, 0, 3], [0, 0, 0, 4, 0, 4], [0, 0, 0, 6, 0, 6], [0, 0, 0, 0, 0, 0]}

10. {[0, 0, 0, 0, 0, 0]}

Here are 48 idempotents of Q(Idl(Z12)) as returned by calling the function idempotents endIdlZ 12 [0, 1, 1, 1, 1, 1], [0, 3, 1, 3, 1, 1], [0, 6, 1, 1, 1, 1], [0, 6, 1, 3, 1, 1], [0, 6, 1, 6, 1, 1], [0, 0, 1, 1, 1, 1], [0, 0, 1, 3, 1, 1], [0, 0, 1, 0, 1, 1], [0, 2, 2, 1, 2, 1], [0, 2, 2, 2, 2, 2], [0, 6, 2, 1, 2, 1], [0, 6, 2, 2, 2, 2], [0, 6, 2, 3, 2, 1], [0, 6, 2, 6, 2, 2], [0, 0, 2, 1, 2, 1], [0, 0, 2, 2, 2, 2], [0, 0, 2, 3, 2, 1], [0, 0, 2, 0, 2, 2], [0, 3, 3, 3, 3, 3], [0, 6, 3, 3, 3, 3], [0, 0, 3, 3, 3, 3], [0, 1, 4, 1, 1, 1], [0, 2, 4, 1, 2, 1], [0, 2, 4, 2, 2, 2], [0, 3, 4, 3, 1, 1], [0, 4, 4, 1, 4, 1], [0, 4, 4, 4, 4, 4], [0, 6, 4, 1, 2, 1], [0, 6, 4, 2, 2, 2], [0, 6, 4, 3, 2, 1], [0, 6, 4, 6, 2, 2], [0, 0, 4, 1, 4, 1], [0, 0, 4, 3, 4, 1], [0, 0, 4, 4, 4, 4], [0, 0, 4, 0, 4, 4], [0, 6, 6, 1, 6, 1], [0, 6, 6, 3, 6, 3], [0, 6, 6, 6, 6, 6], [0, 1, 0, 1, 1, 1], [0, 2, 0, 1, 2, 1], [0, 2, 0, 2, 2, 2], [0, 3, 0, 3, 3, 3], [0, 6, 0, 1, 6, 1], [0, 6, 0, 3, 6, 3], [0, 6, 0, 6, 6, 6], [0, 0, 0, 1, 0, 1], [0, 0, 0, 3, 0, 3], [0, 0, 0, 0, 0, 0] One can then verify that A = Qa for all ⊥= 6 a ∈ A, B = Q[0, 2, 0, 1, 2, 1] = Q[0, 6, 0, 1, 6, 1] = Q[0, 6, 0, 3, 6, 3], and ﬁnally C = Q[0, 0, 0, 1, 0, 1] = Q[0, 0, 0, 3, 0, 3] by calling cyclic endIdlZ 12 map with the respective homomorphism used for map .

30 A.2 Source Code

-- a collection of functions for computations with sup-lattice endomorphism quantales of Idl(Z_n) -- author: Radek Slesinger, [email protected] module Main where import qualified Data.List as L import qualified Data.Set as S import qualified Data.Map as M type Ideal = Int type IdealLattice = S.Set Ideal type SLMorphism = M.Map Ideal Ideal type SLMorphismLattice = S.Set SLMorphism leqEl :: Ideal -> Ideal -> Bool leqEl a b = if b /= 0 then mod a b == 0 else a == 0 -- comparison of ideals, tests whether b | a joinEl :: Ideal -> Ideal -> Ideal joinEl a b = if a == 0 && b == 0 then 0 else gcd a b -- computes the join of ideals a, b endIdlZ_12 :: SLMorphismLattice endIdlZ_12 = S.fromList [M.fromList [(0,0),(6,f6),(4,f4),(3,f3), (2,joinEl f4 f6),(1,joinEl f4 f3)] | f4 <- m, f6 <- m, f3 <- m, leqEl f6 f3] where m = [1,2,3,4,6,0] -- based on admissible images of supercompact elements, generates all sup-lattice endomorphisms for Idl(Z_12) endIdlZ_60 :: SLMorphismLattice endIdlZ_60 = S.fromList [M.fromList [(0,0),(20,f20),(30,f30), (12,f12),(15,f15),(10,joinEl f20 f30), (4,joinEl f20 f12),(6,joinEl f30 f12), (5,joinEl f20 f15),(2,joinEl (joinEl f20 f30) f12), (3,joinEl f12 f15),(1,joinEl (joinEl f12 f15) f20)]| f20 <- m,f30 <- m,f12 <- m,f15 <- m,leqEl f30 f15] where m = [1,2,3,4,5,6,10,12,15,20,30,0] -- based on admissible images of supercompact elements, generates all sup-lattice endomorphisms for Idl(Z_60) leqMor :: SLMorphism -> SLMorphism -> Bool leqMor f g = L.foldr1 (&&) (L.zipWith (\a b -> leqEl a b) (M.elems f) (M.elems g)) -- pointwise comparison of maps, returns True if f is less than g in all elements nleqMor :: SLMorphism -> SLMorphism -> Bool nleqMor f g = not (leqMor f g) -- negation of previous

31 joinMor :: SLMorphism -> SLMorphism -> SLMorphism joinMor = M.unionWith joinEl -- computes pointwise join of two maps isJoinPrime :: SLMorphismLattice -> SLMorphism -> Bool isJoinPrime l f = L.foldr (&&) True [nleqMor f (joinMor g h) | g <- S.elems fl, h <- S.elems fl] where fl = S.filter (nleqMor f) l -- tests whether f is a join-prime in l -- for any g,h from l, f is below g, below h, or not below the join of g and h joinPrimes :: SLMorphismLattice -> S.Set SLMorphism joinPrimes l = S.filter (isJoinPrime l) l -- lists all join-primes in l apply :: SLMorphism -> Ideal -> Ideal apply f x = M.findWithDefault 0 x f -- computes f(x); if f is not defined in x, returns 0 compose :: SLMorphism -> SLMorphism -> SLMorphism compose f g = M.map (apply f) g -- computes the map "f after g" isIdempotent :: SLMorphism -> Bool isIdempotent f = compose f f == f -- tests whether f is an idempotent map idempotents :: S.Set SLMorphism -> S.Set SLMorphism idempotents = S.filter isIdempotent -- lists all idempotent maps in the given set cyclic :: SLMorphismLattice -> SLMorphism -> SLMorphismLattice cyclic l f = S.map (flip compose f) l -- computes the cyclic submodule of l consisting of all elements xf, x from l dsPrinc :: SLMorphismLattice -> SLMorphism -> S.Set SLMorphism dsPrinc l x = S.filter (flip leqMor x) l -- computes the principal downset of x in l: selects all elements of l that are less or equal to f ds :: SLMorphismLattice -> S.Set SLMorphism -> S.Set SLMorphism ds l a = S.unions (S.elems (S.map (dsPrinc l) a)) -- computes the downset of a set a in l -- takes the union of downsets of elements of a

-- demo program: -- computations for endIdlZ_12 take only a few seconds; for endIdlZ_60 half an hour was needed on a C2D 1.83GHz CPU -- for better experience, pretty-printing functions should be added q = endIdlZ_12 main = do

32 putStrLn "join-primes:" putStrLn (show (joinPrimes q)) putStrLn "" putStrLn "down-closed submodules generated by join-primes:" putStrLn (show (S.map(\x->(x,ds q(cyclic q x)))(joinPrimes q))) putStrLn "" putStrLn "idempotents:" putStrLn (show (idempotents q)) putStrLn "" putStrLn "cyclic submodules generated by idempotents:" putStrLn (show (S.map(\x->(x,cyclic q x))(idempotents q)))

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