DOCSLIB.ORG

Semigroups of Terms, Tree Languages, Menger Algebra of N-Ary Functions and Their Embedding Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Semigroups of Terms, Tree Languages, Menger Algebra of N-Ary Functions and Their Embedding Theorems S S symmetry Article Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems Thodsaporn Kumduang 1 and Sorasak Leeratanavalee 1,2,* 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand; [email protected] 2 Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand * Correspondence: [email protected] Abstract: The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we es- tablish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups. Citation: Kumduang, T.; Leeratanavalee, S. Semigroups of Keywords: terms; tree languages; generalized hypersubstitutions; n-ary functions Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their MSC: 08A02; 08A40; 08A70; 20N15 Embedding Theorems. Symmetry 2021, 13, 558. https://doi.org/ 10.3390/sym13040558 1. Introduction and Preliminaries Academic Editor: Ivan Chajda In the classical theory of theoretical computer science, an automaton is a finite state machine which accepts certain strings of letters from a fixed base alphabet. Such strings Received: 3 March 2021 are usually called words, and sets of words are called languages. Formal language theory Accepted: 24 March 2021 Published: 27 March 2021 is the study of properties of languages and automata. One of a generalized study of formal languages is the study of tree languages, i.e., the words of the classical case can be considered as a particular kind of terms. Since terms are commonly represented by Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in tree diagrams, such formal languages are also called tree languages. For more details in published maps and institutional affil- this background, see [1–3]. We now recall the fundamental notion of terms as follows: Let iations. Xn = fx1, ... , xng, for n a natural number, be a finite set which elements are called variables and X = fx1, ... , xn, ...g be countably infinite. The variable xi in Xn is an alphabet of formal languages. To define terms from this alphabet, we use a set f fi j i 2 Ig of operation symbols, indexed by the set I. The type is the sequence t = (ni)i2I of the natural numbers which are arities of the operation symbols f . An n-ary term of type t is defined inductively Copyright: © 2021 by the authors. i by: Every variable x 2 X is an n-ary term of type t and f (t , ... , t ) is an n-ary term Licensee MDPI, Basel, Switzerland. j n i 1 ni This article is an open access article of type t where t1, ... , tni are n-ary terms of type t and fi is an ni-ary operation symbol. S + distributed under the terms and The set of all n-ary terms of type t is denoted by Wt(Xn). Let Wt(X) := n2N Wt(Xn) conditions of the Creative Commons be the set of all terms of type t. See [4–10] for other related topics of terms. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Symmetry 2021, 13, 558. https://doi.org/10.3390/sym13040558 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 558 2 of 13 Every term can be represented by a tree diagram, i.e., we consider term as a rooted tree whose vertices correspond to the operation symbols and variables that occur in it. We call this tree the tree representation of a term t. For instance, a tree representation of a term f (x4, g(h(x1, x2)), h(x9, x5)) can be shown in the below diagram. x1 x2 x9 x5 h x4 g h f Substituting the variables that appear in a term by the other terms, one obtains a new term. This can be described by the (n + 1)-generalized superpostion Sn, n ≥ 1, n n+1 S : Wt(X) ! Wt(X) defined inductively by the following steps: for t, t1,..., tn 2 Wt(X) n (1) If t = xi; 1 ≤ i ≤ n, then S (xi, t1,..., tn) := ti. n (2) If t = xi; n < i, then S (xi, t1,..., tn) := xi. (3) If t = fi(s1,..., sni ), then n n n S (t, t1,..., tn) := fi(S (s1, t1,..., tn),..., S (sni , t1,..., tn)). n Then, we can form the algebraic structure, (n + 1)-ary algebra (Wt(X), S ) consisting n the universe Wt(X) together with one (n + 1)-ary operation S . In [11], the algebra in this form is known as a Menger algebra with infinitely many nullary operations. It is not hard to show that this algebra satisfies the superassociative law: n n n n n S (S (t, t1,..., tn), s1,..., sn) = S (t, S (t1, s1,..., sn),..., S (tn, s1,..., sn)) for all arbitrary terms t, t1,..., tn, s1,..., sn 2 Wt(X). Using the concepts of terms and generalized superposition, the following ideas are es- sentially recalled. It was known from [12] that a formal definition of a strong hyperidentity and a strong solid variety can be given using the concept of a generalized hypersubstitution. We now recall such concept as follows: Let f fi j i 2 Ig be an indexed set of operation symbols of type t where fi is ni-ary, ni is a natural number. Let HypG(t) be the set of all arbitrary mappings s : f fi j i 2 Ig ! Wt(X), which is called a generalized hypersubstitu- tion of type t. To define a binary operation on this set, the essential defining is necessary. Any s 2 HypG(t) can be uniquely extended to a mapping bs : Wt(X) ! Wt(X), which is defined by (1) bs[xi] := xi 2 X, [ ( )] = ni ( ( ) [ ] [ ]) [ ] ≤ ≤ (2) bs fi t1, ... , tni : S s fi , bs t1 , ... , bs tni where bs tj , 1 j ni are already defined. Using the extension of generalized hypersubstitution of type t, the binary operation ◦G is defined by s1 ◦G s2 := bs1 ◦ s2 where ◦ denotes the usual composition of mappings. The generalized hypersubstitution sid, which sends each fi to the term fi(x1, x2, ... , xni ), is an identity element for ◦G. Then, (HypG(t), ◦G, sid) is a monoid. See the following references for the research topics and current trends in this direction [13–18]. Symmetry 2021, 13, 558 3 of 13 Let P(Wt(X)) be the power set of all the set of all terms of type t. An inductive definition of an (n + 1)-ary operation on P(Wt(X)) is completely defined in [19]. Let n be a natural number, B, B1, ... , Bn are arbitrary subsets of Wt(X). Then, an (n + 1)-ary generalized superposition operation n n+1 Sb : P(Wt(X)) ! P(Wt(X)) is defined inductively by n (1) If B = fxig; 1 ≤ i ≤ n, then Sb (fxig, B1,..., Bn) := Bi. n (2) If B = fxig; n < i, then Sb (fxig, B1,..., Bn) := fxig. n (3) If B = f fi(t1, ... , tni )g, and suppose that Sb (ftkg, B1, ... , Bn) for all k = 1, ... , ni n are already defined, then Sb (f fi(t1, ... , tni )g, B1, ... , Bn) := f fi(r1, ... , rni ) j rk 2 n Sb (ftkg, B1,..., Bn), 1 ≤ k ≤ nig. (4) If B is an arbitrary nonsingleton subset of Wt(X), then n [ n Sb (B, B1,..., Bn) := Sb (fbg, B1,..., Bn). b2B n If one of the sets B, B1, ... , Bn is empty, we define Sb (B, B1, ... , Bn) := Æ. It turns out n that the following algebra of type (n + 1), (P(Wt(X)), Sb ) is a Menger algebra and called a power Menger algebra with infinitely many nullary operations. Then, it was proved that this algebra satisfies the superassociative law [19]. A non-deterministic generalized hypersubstitution of type t [19] is a mapping s : nd f fi j i 2 Ig ! P(Wt(X)). The set of all such mappings is denoted by HypG (t). It is well-known that every s generates a mapping bs which takes a tree language into itself by the following inductive way: (1) bs[Æ] := Æ, (2) bs[fxig] := fxig where xi is a variable from X, [f ( )g] = ni ( ( ) [f g] [f g]) [f g] ≤ ≤ (3) bs fi t1, ... , tni : Sb s fi , bs t1 , ... , bs tni if bs tk , 1 k ni are already defined, S (4) bs[B] := bs[fbg] if B is an arbitrary nonsingleton subset of Wt(X). b2B The algebraic properties of bs were proved in [19] that every extension bs of non-deterministic n generalized hypersubstitution s is an endomorphism of the algebra (P(Wt(X)), Sb ). Applying nd nd 2 nd an extension bs, in [20], the binary operation ◦G : HypG (t) ! HypG (t) was introduced nd by setting s1 ◦G s2 := bs1 ◦ s2. Moreover, the non-deterministic generalized hypersubsti- tution sid was defined to be an identity element where sid( fi) := f fi(x1, ..
Load more

Recommended publications
  • Linear Induction Algebra and a Normal Form for Linear Operators Laurent Poinsot
    Linear induction algebra and a normal form for linear operators Laurent Poinsot To cite this version: Laurent Poinsot. Linear induction algebra and a normal form for linear operators. 2013. hal- 00821596 HAL Id: hal-00821596 https://hal.archives-ouvertes.fr/hal-00821596 Preprint submitted on 10 May 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Linear induction algebra and a normal form for linear operators Laurent Poinsot Universit´eParis 13, Sorbonne Paris Cit´e, LIPN, CNRS (UMR 7030), France, [email protected], http://lipn.univ-paris13.fr/~poinsot/ Abstract. The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the latter structure being a consequence of the former. In this contribution, we study the corresponding structure in the linear setting, i.e. in the category of modules over a commutative ring rather than in the category of sets, namely the free module generated by the integers. It also provides free structures of induction algebra and of monoid (in the category of modules).
    [Show full text]
  • A Guide to Self-Distributive Quasigroups, Or Latin Quandles
    A GUIDE TO SELF-DISTRIBUTIVE QUASIGROUPS, OR LATIN QUANDLES DAVID STANOVSKY´ Abstract. We present an overview of the theory of self-distributive quasigroups, both in the two- sided and one-sided cases, and relate the older results to the modern theory of quandles, to which self-distributive quasigroups are a special case. Most attention is paid to the representation results (loop isotopy, linear representation, homogeneous representation), as the main tool to investigate self-distributive quasigroups. 1. Introduction 1.1. The origins of self-distributivity. Self-distributivity is such a natural concept: given a binary operation on a set A, fix one parameter, say the left one, and consider the mappings ∗ La(x) = a x, called left translations. If all such mappings are endomorphisms of the algebraic structure (A,∗ ), the operation is called left self-distributive (the prefix self- is usually omitted). Equationally,∗ the property says a (x y) = (a x) (a y) ∗ ∗ ∗ ∗ ∗ for every a, x, y A, and we see that distributes over itself. Self-distributivity∈ was pinpointed already∗ in the late 19th century works of logicians Peirce and Schr¨oder [69, 76], and ever since, it keeps appearing in a natural way throughout mathematics, perhaps most notably in low dimensional topology (knot and braid invariants) [12, 15, 63], in the theory of symmetric spaces [57] and in set theory (Laver’s groupoids of elementary embeddings) [15]. Recently, Moskovich expressed an interesting statement on his blog [60] that while associativity caters to the classical world of space and time, distributivity is, perhaps, the setting for the emerging world of information.
    [Show full text]
  • Semilattice Sums of Algebras and Mal'tsev Products of Varieties
    Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S.
    [Show full text]
  • UNIVERSAL ALGEBRA Abstract Approved _____MMEMIIIMI Norman Franzen
    AN ABSTRACT OF THE THESIS OF JUNPEI SEKINO for the MASTER OF SCIENCE (Name) (Degree) ,, in MATHEMATICS presented on () t ,{, i >_: ¡ ``¡2 (Major) (Date) Title: UNIVERSAL ALGEBRA Abstract approved _____MMEMIIIMI Norman Franzen In this paper, we are concerned with the very general notion of a universal algebra. A universal algebra essentially consists of a set A together with a possibly infinite set of finitary operations on. A. Generally, these operations are related by means of equations, yield- ing different algebraic structures such as groups, groups with oper- ators, modules, rings, lattices, etc. This theory is concerned with those theorems which are common to all these various algebraic sys- tems. In particular, the fundamental isomorphism and homomorphism theorems are covered, as well as, the Jordan- Holder theorem and the Zassenhaus lemma. Furthermore, new existence proofs are given for sums and free algebras in any primitive class of universal algebras. The last part treats the theory of groups with multi- operators in a manner essentially different from that of P. J. Higgins. The ap- proach taken here generalizes the theorems on groups with operators as found in Jacobson's "Lectures in Abstract Algebra, " vol. I. The basic language of category theory is used whenever conven- ient. Universal Algebra by Junpei Sekino A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science June 1969 APPROVED: Assistant Professor of Mathematics in charge of major Acting Chairman of Department of Mathematics Dean of ra uate c ool Date thesis is presented ( ;( Typed by Clover Redfern for JUNPEI SEKINO ACKNOWLEDGMENT I wish to acknlowledge my gratitude to Mr.
    [Show full text]
  • Universal Algebra Today Part I
    Universal Algebra Today Part I Libor Barto Charles University, Prague 55th Summer School on Algebra and Ordered Sets Nov´ySmokovec, 3{8 Sep 2017 L. Barto (CUNI) UA Today I SSAOS 2017 1 / 26 Outline I Today: Universal Algebra yesterday I Tommorow: Universal Algebra today I Thursday: Universal Algebra tommorow L. Barto (CUNI) UA Today I SSAOS 2017 2 / 26 About the tutorial I I will give you my (present) opinions on I What is Universal Algebra =UA (Part I) I Basic concepts and ideas in UA (Part I) I More advanced concepts and ideas (Part II) I Directions (Part III) I I will concentrate on concepts and proofs. This is not a survey of the most important results. I Interrupt me! I Apologies: typos, ugly slides, incorrect theorems and proofs, . L. Barto (CUNI) UA Today I SSAOS 2017 3 / 26 UA = study of general algebraic structures I Algebras in mathematics I Classic algebras { fields, rings, modules (geometry, analysis, number theory) I Groups as symmetries (algebra, geometry, combinatorics) I Lattices (combinatorics, logic, semantics in CS) I Semigroups (combinatorics, automata and languages) I Quasigroups, . (combinatorics, geometry) I GM: What do you do in UA? I UA: Generalize (HSP, iso theorems, decompositions) I UA: Organize (Mal'tsev conditions) I UA: Study particular classes above { not this tutorial I UA: Develop complicated, monumental, deep, great theories for large classes (commutator theory, TCT) I GM: Why? L. Barto (CUNI) UA Today I SSAOS 2017 4 / 26 Why? I GM: Why do you develop general theories? I UA: To answer complicated questions . in UA For some reason, we are especially excited about identities = universally quantified equations (GM: \hmm, interesting .
    [Show full text]
  • Lectures on Universal Algebra
    Lectures on Universal Algebra Matt Valeriote McMaster University November 8, 1999 1 Algebras In this ¯rst section we will consider some common features of familiar alge- braic structures such as groups, rings, lattices, and boolean algebras to arrive at a de¯nition of a general algebraic structure. Recall that a group G consists of a nonempty set G, along with a binary ¡1 operation ¢ : G ! G, a unary operation : G ! G, and a constant 1G such that ² x ¢ (y ¢ z) = (x ¢ y) ¢ z for all x, y, z 2 G, ¡1 ¡1 ² x ¢ x = 1G and x ¢ x = 1G for all x 2 G, ² 1G ¢ x = x and x ¢ 1G = x for all x 2 G. A group is abelian if it additionally satis¯es: x ¢ y = y ¢ x for all x, y 2 G. A ring R is a nonempty set R along with binary operations +, ¢, a unary operation ¡, and constants 0R and 1R which satisfy ² R, along with +, ¡, and 0R is an abelian group. ² x ¢ (y ¢ z) = (x ¢ y) ¢ z for all x, y, z 2 G. ² x ¢ (y + z) = (x ¢ y) + (x ¢ z) and (y + z) ¢ x = (y ¢ x) + (z ¢ x) for all x, y, z 2 G. 1 ² 1G ¢ x = x and x ¢ 1G = x for all x 2 G. Lattices are algebras of a di®erent nature, they are essentially an algebraic encoding of partially ordered sets which have the property that any pair of elements of the ordered set has a least upper bound and a greatest lower bound. A lattice L consists of a nonempty set L, equipped with two binary operations ^ and _ which satisfy: ² x ^ x = x and x _ x = x for all x 2 L, ² x ^ y = y ^ x and x _ y = y _ x for all x, y 2 L, ² x ^ (y _ x) = x and x _ (y ^ x) = x for all x, y 2 L.
    [Show full text]
  • Universal Algebra and Diagrammatic Reasoning
    Universal Algebra and Diagrammatic Reasoning John Baez Geometry of Computation 2006 January 30 - Feburary 3 = figures by Aaron D. Lauda available online at http://math.ucr.edu/home/baez/universal/ 1 Universal Algebra Modern universal algebra is the study of general mathematical structures, especially those with an `algebraic' flavor. For example: Monads describe `extra algebraic • structure on objects of one fixed category'. PROs describe `extra algebraic • structure on objects of any monoidal category'. PROPs describe `extra algebraic • structure on objects of any sym- metric monoidal category'. Algebraic Theories describe `ex- • tra algebraic structure on objects of any category with finite prod- ucts'. 2 For example: There's a monad on Set whose • algebras are groups. There's an algebraic theory whose • algebras in any category with finite products C are `groups in C'. There's no PROP whose algebras • are groups, but there's a PROP for monoids. PROPs describe fewer structures, but apply to more contexts: e.g., the category of Hilbert spaces with its tensor product. In this quantum- mechanical context we cannot `du- plicate or delete information', so the group axiom g g 1 = 1 · − cannot be expressed. 3 In modern universal algebra we de- scribe operations using diagrams with inputs and outputs. Physicists do the same with Feynman diagrams: ' - ' J - ' K - ÙI ' #c -- ÙI ' K -- ÙI '' _ J ×G - ÙI '' K -ÙI ' J ×G - ÙI ' '' J V ' K Z ' J Q Y Y + U + M + + Y T + O J ^ ^ O ÙI ×G ^ J ÔD ÓC · · · ÕE · · · For example, all the diagrams above stand for operators between Hilbert spaces: He Hγ He ⊗ ! Similar diagrams show up in other contexts: electrical circuits • logic circuits • flow charts • and part of our job is to unify these.
    [Show full text]
  • Universal Algebra and Lattice Theory Week 1 Examples of Algebras
    Universal algebra and lattice theory Week 1 Examples of algebras Charlotte Aten 2020 September 3 . Today’s topics Quick review of the definition of an algebra Magmas Semigroups Monoids Groups Rings Modules Quasigroups Semilattices Lattices n-ary magmas . Definition of an algebra Operations are rules for combining elements of a set together to obtain another element of the same set. Definition (Operation, arity) Given a set A and some n 2 W we refer to a function f : An ! A as an n-ary operation on A. When f is an n-ary operation on A we say that f has arity n. Algebras are sets with an indexed sequence of operations. Definition (Algebra) f g An algebra (A; F) consists of a set A and a sequence F = fi i2I of operations on A, indexed by some set I . Magmas An algebra A := (A; f ) with a single binary operation is called a magma. This is the Bourbaki terminology. These algebras are also known as groupoids and binars, but the term «groupoid» has also become attached to a different concept in category theory. When the set A is finite we can represent the basic operation f : A2 ! A as a finite table, called a Cayley table or operation table for f . Magmas f r p s r r p r p p p s s r s s Figure: A Cayley table for a binary operation f The above table defines a binary operation f : A2 ! A where A := fr; p; sg. For example, f (r; p) = p. The magma A := (A; f ) is the rock-paper-scissors magma.
    [Show full text]
  • Deductive Varieties of Modules and Universal Algebras by Leslie Hogben and Clifford Bergman1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 289, Number 1. May 1985 DEDUCTIVE VARIETIES OF MODULES AND UNIVERSAL ALGEBRAS BY LESLIE HOGBEN AND CLIFFORD BERGMAN1 Abstract. A variety of universal algebras is called deductive if every subquasi- variety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above. This paper initiates a study of varieties of universal algebras with the property that every subquasivariety is a variety. We call such varieties deductive. In §2 it is proved that the variety of left modules of a left Artinian ring R is deductive if and only if R is the direct sum of matrix rings over local rings in which the maximal ideal is principal as a left and right ideal. In §3 we consider varieties of universal algebras of finite type that are directly representable. A variety is directly representable if it is generated by a finite algebra and has, up to isomorphism, only finitely many directly indecomposable finite algebras. Such a variety is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element and a certain ring, constructed from the variety, has the properties listed above.
    [Show full text]
  • Introducing Boolean Semilattices
    Introducing Boolean Semilattices Clifford Bergman The study of Boolean algebras with operators (BAOs) has been a consistent theme in algebraic logic throughout its history. It provides a unifying framework for several branches of logic including relation algebras, cylindric algebras, and modal alge- bras. From a purely algebraic standpoint, a class of BAOs provides a rich field of study, combining the strength of Boolean algebras with whatever structure is im- posed on the operators. In fact, with all this structure, one might expect that analyzing a variety of BAOs would border on the trivial. The variety of Boolean algebras, after all, is generated by a primal algebra. As such, it is congruence-distributive, congruence-permutable, semisimple, equationally complete, has EDPC,. , the list goes on. And yet it turns out that for all but the most degenerate operators, the analysis is anything but simple. The explanation is in the intricate and unexpected interplay between the Boolean operations and the additional operators that arise from standard constructions. In this paper we consider Boolean algebras with one very simple operator, namely an (almost) semilattice operation, that is, a binary operation that is asso- ciative, commutative, and (almost) idempotent. The qualification on idempotence will be explained below. We shall develop some of the arithmetic of these algebras, discuss some representation questions, and pose some problems. While there are no deep results in this work, we hope that it will stimulate further research in this interesting class of algebras. Peter Jipsen is responsible, for better or worse, for introducing me to represen- tation questions for BAOs.
    [Show full text]
  • A Course in Universal Algebra
    The Millennium Edition Stanley Burris H. P. Sankappanavar A Course in Universal Algebra With 36 Illustrations c S. Burris and H.P. Sankappanavar All Rights Reserved This Edition may be copied for personal use This book is dedicated to our children Kurosh Phillip Burris Veena and Geeta and Sanjay Sankappanavar Preface to the Millennium Edition The original 1981 edition of A Course in Universal Algebra has now been LaTeXed so the authors could make the out-of-print Springer-Verlag Gradu- ate Texts in Mathematics edition available once again, with corrections. The subject of Universal Algebra has flourished mightily since 1981, and we still believe that A Course in Universal Algebra offers an excellent introduction to the subject. Special thanks go to Lis D’Alessio for the superb job of LaTeXing this edition, and to NSERC for their support which has made this work possible. v Acknowledgments First we would like to express gratitude to our colleagues who have added so much vital- ity to the subject of Universal Algebra during the past twenty years. One of the original reasons for writing this book was to make readily available the beautiful work on sheaves and discriminator varieties which we had learned from, and later developed with H. Werner. Recent work of, and with, R. McKenzie on structure and decidability theory has added to our excitement, and conviction, concerning the directions emphasized in this book. In the late stages of polishing the manuscript we received valuable suggestions from M. Valeriote, W. Taylor, and the reviewer for Springer-Verlag. For help in proof-reading we also thank A.
    [Show full text]
  • Quasigroups and Quandles
    Discrete Mathematics 109 (1992) 277-282 277 North-Holland Quasigroups and quandles Jonathan D.H. Smith Department of Mathematics. Iowa State University, Ames, IA _Wll, USA Received March 1990 Revised 1 March 1991 Dedicated to Prof. Gert Sabidussi on the occasion of his sixtieth birthdav. Abstract Smith, J.D.H., Quasigroups and quandles, Discrete Mathematics 109 (1992) 277-282. Two connexions between quasigroups and quandles are established. In one direction, Joyce’s homogeneous quandle construction is shown to yield a quasigroup isotopic to the loop constructed by Scimemi on the set of @-commutators of a group automorphism @ In the other direction, the universal multiplication group construction of quasigroup theory is extended to quandles. The group of a knot becomes the universal right multiplication group of the knot quandle. 0. Introduction A q~~~igr~~~p (Q, -) is a set Q equipped with a binary multiplication - such that. in the equation x-y=%, (0. I) knowledge of any two of x, y, z specifies the third uniquely. It follows that for each element x of Q, the right multiplication R(x):Q-+Q;y-y-x (O-2) and left mi.dtiplicatioYt L(x):Q+Q;y-x-y (0.3) are permutations of Q. From the standpoint of universal algebra, the definition (0.1) is rather awkward, since it (implicitly) involves quantifiers. An equivalent [8, p. 1171 definition of a quasigroup (QP -, /, \) may be given as a set Q equipped with three binary operations, namely muftipiication, right division, and left division respectively, such that the identities (ER): (x/y) - y = x, (0.4) (UR): (x - y)/y =x 0612~365X/92;$05.00 @ 1992- Elsevier Science Publishers B.V.
    [Show full text]