DOCSLIB.ORG

Groupoids on a Skew Lattice of Objects

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Groupoids on a Skew Lattice of Objects ISSN 2590-9770 The Art of Discrete and Applied Mathematics 2 (2019) #P2.03 https://doi.org/10.26493/2590-9770.1342.109 (Also available at http://adam-journal.eu) Groupoids on a skew lattice of objects Desmond G. FitzGerald ∗ School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart 7001, Australia To Jonathan Leech—Na zdravje! Received 16 October 2018, accepted 11 December 2019, published online 27 December 2019 Abstract Motivated by some alternatives to the classical logical model of boolean algebra, this paper deals with algebraic structures which extend skew lattices by locally invertible ele- ments. Following the meme of the Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of isomorphisms) in which the set of objects carries the struc- ture of a skew lattice. The objects act on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Condi- tions are placed on the actions, from which pseudoproducts may be defined. This gives an algebra of signature (2; 2; 1), in which each binary operation has the structure of an orthodox semigroup. In the reverse direction, a groupoid of the kind described may be reconstructed from the algebra. Keywords: Inductive groupoids, skew lattices, orthodox semigroups. Math. Subj. Class.: 20L05, 20M19, 06B75 1 Non-commutative and non-idempotent lattice analogues As non-classical logics have been developed for various knowledge domains, so various algebras have been proposed as extensions or alternatives to the classical model of boolean algebra. A significant one for this paper is the theory of skew lattices; for a contemporary account, see Leech’s surveys [10, 11]. We provide the details of our notation in Section3. Another proposal is that of MV-algebras, and their coordinatisation¨ via inverse semigroups as described by Lawson and Scott [9]. Thus one theme is to allow sequential operations and hence non-commutative logical connectives, and another introduces non-idempotent ∗The author thanks Vicky Gould, Michael Kinyon and Anja Kudryavtseva for their interest and input in discus- sions, both in person and by email, which helped convert a half-baked workshop idea into an interesting project. E-mail address: d.fi[email protected] (Desmond G. FitzGerald) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 2 (2019) #P2.03 connectives. This note will consider a combination of these themes, by seeking reasonable structures which extend skew lattices by locally invertible elements. The principal tool in this construction is based on the ideas behind the Ehresmann- Schein-Nambooripad (ESN) theorem, of which a full account is given in Chapter 4 of Lawson’s book [8]: we consider a small category of isomorphisms in which the set of objects carries the structure of a skew lattice. We postulate that the objects act (partially) on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Certain reasonable conditions are postulated, and from these a suitable pseudoproduct is defined, much as in the inverse semigroup case, for each skew lattice operation (a non-commutative “meet” and “join”). This results in a total algebra involving two orthodox semigroups with a common set of idempotents isomorphic to the given skew lattice. Because of the complexity involved in having two operations, we begin by considering a groupoid over a set of objects with a single band operation. A much more general situation has been studied, under the name of weakly B-orthodox semigroups, by Gould and Wang in [2], but because the present author has been unable to find this special case treated in the literature, a detailed account will be given here. Later sections deal with the pair of linked band operations, construct the total algebra described above, and show how the original groupoid may be recovered from the algebra. Aspects of the constructions which need further elaboration are noted in the final section. 2 Groupoids on a band of objects Let us recall from [8] that an inverse semigroup is equivalent to an inductive groupoid, i.e., • a (small) category of isomorphisms with • a meet operation on objects and • a notion of restriction of a morphism to any of its subdomains. We attempt something similar here, but changing the conditions on the set of objects. Let G be a groupoid with composition ◦ and B its set of objects, endowed with an associa- tive and idempotent operation ^. Then (B; ^) is known as a lower band, and possesses a pair of natural preorders: we write • a ≤L b if and only if a = a ^ b, and a ≤R b if and only if a = b ^ a. As usual, we may identify each object b with its identity ib, and write dg and rg for the domain and range maps in G , thus: dg = g ◦ g−1; rg = g−1 ◦ g. Suppose too that for each a 2 B there are left and right restriction (partial) operations aj; ja : G ! G such that: • ajg is defined whenever a ≤L dg, with ajg : a ! r(ajg) ≤L rg; and (lateral-) dually, • gja is defined whenever a ≤R rg, with gja : d(gja) ! a; d(gja) ≤L dg. Figure1 shows the left and right restrictions. (There are analogous (in fact, vertically dual) requirements for extension operators, which will be dealt with more explicitly in Section3.) Certain sensible axioms must be satisfied: D. G. FitzGerald: Groupoids on a skew lattice of objects 3 g g c d c d a ≤L c a ≤R d a r(ajg) d(gja) a ajg gja Figure 1: Left and right restriction operators. (i) (identities) d(g)jg = g; (ii) (preorders) if a ≤L b, then ajib = ia; (iii) (transitivity) if a ≤L b ≤L dg, then ajg = a^bjg = aj(bjg); (iv) (composition) if f ◦ g is defined (so that rf = dg), then aj(f ◦ g) = (ajf) ◦ (r(ajf)jg); the right-hand composite being defined because r(ajf) ≤L rf = dg; see Figure2. f g b c d ≤L ≤L ajf r(ajf)jg a r(ajf) r(ajf ◦ g) aj(f ◦ g) Figure 2: Restriction of a composite morphism. 2.1 Actions and conjugacy g g Let us write, without prejudice, a as an alternative for r(ajg), and a for d(gja). This lightens the notation, and emphasises the similarity to actions and conjugates. Caution: f −1 However, it should not be taken to mean that anything like a = (f ja) ◦ (ajf), or f f −1 a = a, or ajf = fjaf necessarily hold: in general, fjaf = fjrf^af . What we do have, following from aj(f ◦ g) = (ajf) ◦ (r(ajf)jg), is that −1 ajib = ia = (ajf) ◦ (af jf ); 4 Art Discrete Appl. Math. 2 (2019) #P2.03 so −1 −1 ib f −1 (ajf) = af jf ; a = a; and a = (af jf ) ◦ (ajf); moreover, af◦g = (af )g. We want to link right and left “actions” by (ajf)jb = aj(fjb), but there is a little problem here, since one of the two sides of that equation may fail to be defined while the other is defined. We therefore seek to extend the conditionally-defined restrictions to total maps by the following device, based on the pseudoproduct construction familiar from the ESN theorem: For g 2 G and any a 2 B, define ajg := a^dgjg, the right hand side being meaning- ful since a ^ dg ≤L dg. (Note that if a ≤ dg already, the notations agree.) The next figure shows the situation (where, also by extension, we write ag for the already-defined (a ^ dg)g). g a b c ≤L a ^ b (a ^ b)g = ag ajg Figure 3: Generalised restriction of g to object a and action of g on a. Then if g = ib, we have a ^ ib = a^bjib = ia^b = ia ^ ib = ia ^ b = iaja^b, and we may write a ^ g for ajg without conflict. A little re-writing of definitions shows that (a ^ b) ^ g = a^bjg = aj(bjg) = a ^ (b ^ g) (2.1) and a ^ dg = d(a ^ g) = (a ^ g) ◦ (a ^ g)−1: (2.2) We complete our list of postulates with the previously-mentioned (ajf)jb = aj(fjb), which we now write as • (a ^ f) ^ b = a ^ (f ^ b); for all a; b 2 B and f 2 G . (More fully, this is (a^df jf)jaf ^b = a^f bj(fjb^rf ).) Next, we may extend the composition further, to a pseudoproduct ⊗: when f : z ! a and g : b ! c, we define f ⊗ g := (fja^b) ◦ (a^bjg) = (f ^ (a ^ b)) ◦ ((a ^ b) ^ g): The pseudoproduct is defined for all pairs f; g. Then a ^ f is actually just ia ⊗ f. This is indeed an extension of meaning: when f ◦ g is defined, f ⊗ g = f ◦ g, and when f = ia and g = ib, f ⊗ g = ia ⊗ ib = ia^b = ia ^ ib; so we may as well use just the one symbol ^ for ⊗, as it extends ◦ and the restrictions, as well as the original ^ on B. Let us check remaining non-trivial cases for associativity. D. G. FitzGerald: Groupoids on a skew lattice of objects 5 f g z a b c ≤R ≤L f (a ^ b) a ^ b (a ^ b)g fja^b a^bjg Figure 4: Diagram illustrating the pseudoproduct. Lemma 2.1. For all f; g 2 G and e 2 B, with f : df ! a and g : b ! rg, (i) (f ^ e) ^ g = f ^ (e ^ g), (ii) ef^g = (ef )g, and (iii) ej(f ^ g) = (ejf) ^ g. Proof. (i): By definition, (f ^ e) ^ g = (fje) ^ g = (fja^e)jb ◦ a^ejg = (fja^e^b) ◦ (a^e^bjg); while f ^ (e ^ g) = fje^b ^ e^bjg = f ^ (e ^ g).
Load more

Recommended publications
  • João Pita Costa on the COSET STRUCTURE of a SKEW LATTICE
    DEMONSTRATIO MATHEMATICA Vol.XLIV No4 2011 João Pita Costa ON THE COSET STRUCTURE OF A SKEW LATTICE Abstract. The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of sets where the Green’s relation D is a congruence describing an adjunction to the category of lattices. In this paper we will discuss the rele- vance of this approach, revisit some known decompositions and relate the order structure of a skew lattice with its coset structure that describes the internal coset decomposition of the respective skew lattice. Introduction Skew lattices have been studied for the past thirty years as a noncom- mutative variation of the variety of lattices with motivations in semigroup theory, in linear algebra as well as in universal algebra. The study of non- commutative lattices began in 1949 by Pascual Jordan [14] who in 1961 presented a wide review on the subject [15]. It is later approached by Slavík [26] and Cornish [4] who refer to a special variety of noncommutative lat- tices, namely skew lattices. A more general version of these skew lattices is due to Jonathan Leech, was first announced in [20] and consists of the left version of Slavík’s algebras. The Green’s relation D defined in a skew lattice S is a congruence and has revealed its important role in the study of these algebras, permitting us a further approach to the coset structure of a skew lattice [23]. The first section of this paper is dedicated to the approach to skew lat- tices as algebraic theories discussing several characterizations related to the choice of the possible absorption laws.
    [Show full text]
  • Fuzzy Ideals of Skew Lattices
    BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Bull. Int. Math. Virtual Inst., Vol. 10(3)(2020), 473-482 DOI: 10.7251/BIMVI2003473G Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) FUZZY IDEALS OF SKEW LATTICES Yeshiwas Mebrat Gubena Abstract. In this paper, the concept of fuzzy ideal of a skew lattice S is introduced, a fuzzy ideal of S is characterized in terms of a crisp ideal of S and equivalent conditions between a fuzzy subset of S and a fuzzy ideal of S are established. Also, it is proved that the set of all fuzzy ideals of a strongly distributive skew lattice forms a complete lattice. Lastly a mapping from the set I(S) of all ideals of a skew lattice S into the set F αI (S) of all α−level fuzzy ideals (αI ) of S corresponding to the ideal I is defined and it is proved that this map is an isomorphism. 1. Introduction The concept of fuzzy subsets of a nonempty set S as a function of S into the unit interval [0; 1] introduced by L. Zadch [12] initiated several algebraists to take up the study of fuzzy subalgebras of various algebraic systems such as groups, rings, modules, lattices etc. The study of fuzzy algebraic structure has started by Rosenfeld [11] and since then this concept has been applied to a variety of algebraic structures such as the notion of a fuzzy subgroup of a group [7] and [11], fuzzy subrings [8], fuzzy ideals of rings [7, 9, 10], and fuzzy ideals of lattices [1].
    [Show full text]
  • Skew Lattices
    Proyecciones Journal of Mathematics Vol. 30, No 1, pp. 51-57, May 2011. Universidad Cat´olica del Norte Antofagasta - Chile Skew lattices K. V. R. Srinivas Regency Institute of Technology, Near Kakinada Received : April 2009. Accepted : January 2011 Abstract In this paper mainly important properties of skew lattices and sym- metric Lattices is obtained. A necessary and sufficient condition for skewlatticetobesymmetricisobtained.Maximalelementofaskew lattice is also obtained. Key words : Skew lattice, symmetric lattice. AMS(2000) subject classification : 20 M 10. 52 K. V. R. Srinivas 1. Introduction In general terms , a non-commutative lattice is an algebra (S, ,), where both and are associative ,Idempotent binary operations,connected by absorption laws.Pascal Jordon discussed about non-commutative lattices in 1949, pa- per[4], work within the scope of jordons approach has been carried out by Gerhardts in e.g[1] and [2]. Further the developments in non-commutative lattice is obtained by schein in[5] and are also obtained by schweigert [7] and [8]. In this paper mainly important properties of skew lattices and symmetric lattices is obtained . By a skew lattice is meant an algebra (S,) where S is a non-empty set, both , are binary operations called the join and meet respectively,satisifies the following identities. SL1:(xy)z=x(yz)and(xy)z=x(yz) SL2:xx=xandxx=x SL3:x(xy)=x(xy)and(xy)y=y=(xy)y The identities found in SL1-SL3 are known as the associative law,the idempotent laws and absorption laws respectively. Assuming the pair of laws ,the absorption laws are equivalent to the following pair of absorption equivalences.
    [Show full text]
  • Are a Non-Commutative Version of Lattices
    A NON-COMMUTATIVE PRIESTLEY DUALITY ANDREJ BAUER, KARIN CVETKO-VAH, MAI GEHRKE, SAM VAN GOOL, AND GANNA KUDRYAVTSEVA Abstract. We prove that the category of left-handed skew distributive lat- tices with zero and proper homomorphisms is dually equivalent to a cate- gory of sheaves over local Priestley spaces. Our result thus provides a non- commutative version of classical Priestley duality for distributive lattices. The result also generalizes the recent development of Stone duality for skew Boolean algebras. 1. Introduction Skew lattices [12, 13] are a non-commutative version of lattices: algebraically, a skew lattice is a structure (S; _; ^), where _ and ^ are binary operations which satisfy the associative and idempotent laws, and certain absorption laws (see 2.1 below). Concrete classes of examples of skew lattices occur in many situations. The skew lattices in such classes of examples often have a zero element, and also satisfy certain additional axioms, which are called distributivity and left-handedness (see 2.3 and 2.4 below). A (proto)typical class of such examples is that of skew lattices of partial functions, that we will describe now. If X and Y are sets, then the collection S of partial functions from X to Y carries a natural skew lattice structure, as follows. If f; g 2 S are partial functions, we define f ^ g to be the restriction of f by g, that is, the function with domain dom(f) \ dom(g), where its value is defined to be equal to the value of f. We define f _ g to be the override of f with g, that is, the function with domain dom(f) [ dom(g), where its value is defined to be equal to the value of g whenever g is defined, and to the value of f otherwise.
    [Show full text]
  • Abstracts.Pdf
    The character and the pseudo-character in endomorphism rings Horea Abrudan Technical College ”Mihai Viteazul” Oradea Let A be an Abelian group and End(A) the ring of endomorphisms viewed as a topo- logical ring with the topology of pointwise convergence (finite topology). We estimate the character of End(A) for a torsion group. A characterization of torsion group A for which the pseudo-character is ≤ m, where m is a cardinal number, is given. Quasialgebras, Lattices and Matrices Helena Albuquerque coauthored by Rolf Soeren Kraushnar Departamento de Matemtica- Universidade de Coimbra- Portugal In this talk we will present the notion of quasialgebra and we will give some examples like the octonions. We will study deformed group algebras as Cayley algebras or Clifford algebras and in this context the structure of a lattice defined in the space is induced naturally. On the term functions and fundamental relation of a fuzzy hyperalgebra Reza Ameri coauthored by Tahere Nozari Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran In this paper, we start from this idea that the set of nonzero fuzzy subsets of a fuzzy hyperalgebra can be organized naturally as a universal algebra, and constructing the term functions over this algebra we can deduce some results on fuzzy hyperalgebras from some of the very known properties of the term functions of a universal algebra. Also, we present the form of the generated subfuzzy hyperalgebra in two theorem, which in the particular cases of the universal algebras and multialgebras are already known. In the last section we will present the form of the fundamental relation of a fuzzy hyperalgebra, i.e.
    [Show full text]
  • On Noncommutative Generalisations of Boolean Algebras∗
    ISSN 2590-9770 The Art of Discrete and Applied Mathematics 2 (2019) #P2.07 https://doi.org/10.26493/2590-9770.1323.ceb (Also available at http://adam-journal.eu) On noncommutative generalisations of Boolean algebras∗ Antonio Bucciarelli Institut de Recherche en Informatique Fondamentale, Universite´ de Paris, 8 Place Aurelie´ Nemours, 75205 Paris Cedex 13, France Antonino Salibra y Department of Environmental Sciences, Informatics and Statistics, Universita` Ca’ Foscari Venezia, Via Torino 155, 30173 Venezia, Italia Received 22 August 2019, accepted 7 December 2019, published online 30 December 2019 Abstract Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions. Keywords: Skew Boolean algebras, Boolean-like algebras, Church algebras, multideals. Math. Subj. Class.: 06E75, 03G05, 08B05, 08A30 1 Introduction Boolean algebras are the main example of a well-behaved double-pointed variety – mean- ing a variety V whose type includes two distinct constants 0; 1 in every nontrivial A 2 V. Since there are other double-pointed varieties of algebras that have Boolean-like features, in [15, 23] the notion of Boolean-like algebra (of dimension 2) was introduced as a gen- eralisation of Boolean algebras to a double-pointed but otherwise arbitrary similarity type.
    [Show full text]
  • Filters in Residuated Skew Lattices: Part Ii
    Honam Mathematical J. 40 (2018), No. 3, pp. 401–431 https://doi.org/10.5831/HMJ.2018.40.3.401 (SKEW) FILTERS IN RESIDUATED SKEW LATTICES: PART II R. Koohnavard and A. Borumand Saeid∗ Abstract. In this paper, some kinds of (skew) filters are defined and are studied in residuated skew lattices. Some relations are got between these filters and quotient algebras constructed via these filters. The Green filter is defined which establishes a connection between residuated lattices and residuated skew lattices. It is in- vestigated that relationships between Green filter and other types of filters in residuated skew lattices and the relationship between residuated skew lattice and other skew structures are studied. It is proved that for a residuated skew lattice, skew Hilbert algebra and skew G-algebra are equivalent too. 1. Introduction Skew lattices were introduced for the first time by Jordan [10]. The study of skew lattices began with the 1989 paper of Leech [16]. Skew lattices are a generalization of lattices that operations _; ^ are not com- mutative. In skew lattices, two different order concepts can be defined: the natural preorder, denoted by and the natural partial order denoted by ≤. For given a skew lattice (A; _; ^) both algebraic reducts (A; _) and (A; ^) are bands, that is semigroups whose elements are idempotent (x2 = x). J. Leech explained skew Boolean algebras are non-commutative one- pointed generalisations of (possibly generalised) Boolean algebras [14], each right-handed skew Boolean algebra can be embedded into a generic skew Boolean algebra of partial functions from a given set to the codomain Received October 28, 2017.
    [Show full text]
  • Noncommutative Lattices by Jonathan E. Leech
    Noncommutative Lattices by Jonathan E. Leech J. E. Leech, Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond, volume 4 of Famnit Lectures, Slovenian Discrete and Applied Mathematics Society and University of Primorska Press, Koper, 2021. ISBN 978-961-95273-0-6 (printed edn.), ISBN 978-961-293-027-1 (electronic edn.) Weakenings of lattices, where the meet and join operations may fail to be commutative, attracted from time to time the attention of various communities of scholars, including ordered algebra theorists (for their connection with preordered sets) and semigroup theo- rists (who viewed them as structurally enriched bands possessing a dual multiplication). Recently, noncommutative generalisations of lattices and related structures have seen a growth in interest, with new ideas and applications emerging. The adjective “noncommu- tative” is used here in the inclusive sense of “not-necessarily-commutative”. Much of this recent activity derives in some way from the initiation, thirty years ago, by Jonathan Leech, of a research program into structures based on Pascual Jordan’s notion of a noncommuta- tive lattice. Indeed, the research began by studying multiplicative bands of idempotents in rings, and realising that under certain conditions such bands would also be closed under an “upward multiplication”. In particular, for multiplicative bands that were left regular, any maximal such band in a ring was also closed under the circle operation (or quadratic join) x ◦ y = x + y − xy. And any band closed under both operations satisfied certain absorption identities, e.g., e(e◦f) = e = e◦(ef). These observations indicated the presence of struc- turally strengthened bands with a roughly lattice-like structure.
    [Show full text]
  • Arxiv:1702.04949V2 [Math.RA] 20 Dec 2017 H Edrwoi Neetdi Plctoso Noncommutat of [6]
    NONCOMMUTATIVE FRAMES KARIN CVETKO-VAH Abstract. We explore algebraic properties of noncommutative frames. The concept of noncommutative frame is due to Le Bruyn, who in- troduced it in connection with noncommutative covers of the Connes- Consani arithmetic site. 1. Introduction The motivation for our definition of a noncommutative frame comes from the following interesting example of noncommutative covers on the arith- metic site that is due to Le Bruyn [10]. We refer the reader to [3] and [4] for the definitions of an arithmetic site, and to [9] for the definition of the sieve topology on the arithmetic site. Example 1.1 ([10]). The sieve topology on the arithmetic site is defined by the basic open sets that correspond to sieves S ∈ Ω and are denoted by Xs(S), where Ω is the subobject-classifier of the arithmetic site topos Cˆ. The sheaf Oc of constructible truth fluctuations has as sections over the open set Xs(S) for S ∈ ω the set of all continuous functions x from Xs(S) (with the induced patch topology) to the Boolean semifield B = {0, 1} (with the discrete topology). A noncommutative frame Θ, which represents the set of opens of a noncommutative topology, is defined in [10] as the set of all pairs (S,x), where S ∈ Ω and x : Xs(S) → B continuous. The noncommutative frame operations are defined on Θ by: (S,x) ∧ (T,y) = (S ∧ T,x|Xs(S)∩Xs(T )) (S,x) ∨ (T,y) = (S ∨ T,y|Xs(T ) ∪ x|Xs(S)−Xs(T )) arXiv:1702.04949v2 [math.RA] 20 Dec 2017 (S,x) → (T,y) = (S → T,y ∪ 1Xs(S→T )−Xs(T )).
    [Show full text]
  • Cancellation in Skew Lattices 1
    CANCELLATION IN SKEW LATTICES KARIN CVETKO-VAH, MICHAEL KINYON, JONATHAN LEECH, AND MATTHEW SPINKS Abstract. Distributive lattices are well known to be precisely those lattices that possess cancellation: x _ y = x _ z and x ^ y = x ^ z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the 5-element lattices M3 or N5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ^ and _ no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M3 or N5 that insure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x_z = y_z and x^z = y^z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation. 1. Introduction Recall that a lattice L := hL; ^; _i is distributive if the identity x ^ (y _ z) = (x ^ y) _ (x ^ z) holds on L. One of the first results in lattice theory is the equivalence of this identity to its dual, x_(y^z) = (x_y)^(x_z). Distributive lattices are also characterized as being cancellative: x _ y = x _ z and x ^ y = x ^ z jointly imply y = z. Another characterization is that neither of the following 5-element lattices can be embedded in the given lattice. (The relevant discussion is given in any introduction to lattice theory.
    [Show full text]
  • Boolean Skew Algebras
    Acta Mathematica Academiae Scientiarurn Hungarkcae Tomus 36 (3-----4) (1980), pp. 281--291. BOOLEAN SKEW ALGEBRAS By W. H. CORNISH (Adelaide) 1. Introduction In this paper we introduce a class of skew lattices which generalizes relatively complemented distributive lattices with a smallest element. A member (-4; A, V, 0) of this class can be considered as an algebra of type (2, 2, 0) satisfying the iden- tities: aAa=a, ah(bhe)=(ahb)Ae, aA(bVc)=(ahb)V(aAc), (bVe)Aa= :(bAa)V(cAa), (bAa)Va=a, and the condition: for all a, bCA, there exists c~A such that a=(aAb)Ve and cAb=O. The element c in this last condition is uni- quely determined by a and b and is denoted by r(a, b). In this way, the class gives rise to a variety of algebras (A; A, V, r, 0) of type (2, 2, 2, 0); we call it the variety of Boolean skew algebras. Consequences of our axioms are the identities aV a=a, aV (b V c)=(aV b) V c, a=aV (aAb)=(aAb) V a=aA (aV b)=aA (b V a)=(b V a) A Aa, (bAc)Va:(bVa)A(cVa), aA0=0=0Aa, and aVO=a=OVa. Thus, we reallyare considering a class of skew lattices and it turns out that each of the identities: aAb=bAa, aVb:bVa, a=aV(bAa), a=(aVb)Aa, and aV(bAe):(aVb)A(aVc), defines the same proper subvariety. On any Boolean skew algebra, the maps x-~xVa and x-,-xAa are actually endomorphisms.
    [Show full text]
  • View This Volume's Front and Back Matter
    Other Titles in This Series 43 Lui s A. Caffarelli an d Xavier Cabre, Full y nonlinear elliptic equations, 199 5 42 Victo r duillemin and Shlomo Sternberg, Variation s on a theme by Kepler, 199 0 41 Alfre d Tarski and Steven Gtvant, A formalization o f set theory without variables, 198 7 40 R . H. Bing, Th e geometric topology of 3-manifolds , 198 3 39 N . Jacobson, Structur e and representations of Jordan algebras, 196 8 38 O . Ore, Theor y of graphs, 196 2 37 N . Jacobson, Structur e of rings, 195 6 36 W . H. Gottschalk an d G. A. Hedlund, Topologica l dynamics , 195 5 35 A , C Schaeffe r an d D. C. Spencer, Coefficien t region s for Schlicht functions , 195 0 34 J . L, Walsh, Th e location of critical points of analytic and harmonic functions, 195 0 33 J . F. Ritt, Differentia l algebra , 195 0 32 R . L. Wilder, Topolog y of manifolds, 194 9 31 E . Hille and R. S. Phillips, Functiona l analysis and semigroups, 195 7 30 T . Radd, Lengt h and area, 194 8 29 A . Weil, Foundation s of algebraic geometry, 194 6 28 G . T. Whyburn, Analyti c topology, 194 2 27 S . Lefschetz, Algebrai c topology, 194 2 26 N . Levinson, Ga p and density theorems, 194 0 25 Garret t Birkhoff, Lattic e theory, 194 0 24 A . A. Albert, Structur e of algebras, 193 9 23 G . Szego, Orthogona l polynomials, 193 9 22 C , N. Moore, Summabl e series and convergence factors, 193 8 21 J .
    [Show full text]