Tutorial on Universal Algebra
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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). -
Universal Algebra
Donald Sannella and Andrzej Tarlecki Foundations of Algebraic Specification and Formal Software Development September 29, 2010 Springer Page: v job: root macro: svmono.cls date/time: 29-Sep-2010/17:45 Page: xiv job: root macro: svmono.cls date/time: 29-Sep-2010/17:45 Contents 0 Introduction .................................................... 1 0.1 Modelling software systems as algebras . 1 0.2 Specifications . 5 0.3 Software development . 8 0.4 Generality and abstraction . 10 0.5 Formality . 12 0.6 Outlook . 14 1 Universal algebra .............................................. 15 1.1 Many-sorted sets . 15 1.2 Signatures and algebras . 18 1.3 Homomorphisms and congruences . 22 1.4 Term algebras . 27 1.5 Changing signatures . 32 1.5.1 Signature morphisms . 32 1.5.2 Derived signature morphisms . 36 1.6 Bibliographical remarks . 38 2 Simple equational specifications ................................. 41 2.1 Equations . 41 2.2 Flat specifications . 44 2.3 Theories . 50 2.4 Equational calculus . 54 2.5 Initial models . 58 2.6 Term rewriting . 66 2.7 Fiddling with the definitions . 72 2.7.1 Conditional equations . 72 2.7.2 Reachable semantics . 74 2.7.3 Dealing with partial functions: error algebras . 78 2.7.4 Dealing with partial functions: partial algebras . 84 2.7.5 Partial functions: order-sorted algebras . 87 xv xvi Contents 2.7.6 Other options . 91 2.8 Bibliographical remarks . 93 3 Category theory................................................ 97 3.1 Introducing categories. 99 3.1.1 Categories . 99 3.1.2 Constructing categories . 105 3.1.3 Category-theoretic definitions . 109 3.2 Limits and colimits . 111 3.2.1 Initial and terminal objects . -
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. -
On Birkhoff's Common Abstraction Problem
F. Paoli On Birkho®'s Common C. Tsinakis Abstraction Problem Abstract. In his milestone textbook Lattice Theory, Garrett Birkho® challenged his readers to develop a \common abstraction" that includes Boolean algebras and lattice- ordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of BA and LG their join BA_LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for BA_LG relative to FL. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing BA _ LG. Keywords: Residuated lattice, FL-algebra, Substructural logics, Boolean algebra, Lattice- ordered group, Birkho®'s problem, History of 20th C. algebra. 1. Introduction In his milestone textbook Lattice Theory [2, Problem 108], Garrett Birkho® challenged his readers by suggesting the following project: Develop a common abstraction that includes Boolean algebras (rings) and lattice ordered groups as special cases. Over the subsequent decades, several mathematicians tried their hands at Birkho®'s intriguing problem. Its very formulation, in fact, intrinsically seems to call for reiterated attempts: unlike most problems contained in the book, for which it is manifest what would count as a correct solution, this one is stated in su±ciently vague terms as to leave it open to debate whether any proposed answer is really adequate. It appears to us that Rama Rao puts things right when he remarks [28, p. -
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. -
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. -
How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0. -
From Distributive -Monoids to -Groups, and Back Again
From distributive ℓ-monoids to ℓ-groups, and back again Almudena Colacito1 Laboratoire J. A. Dieudonne,´ Universite´ Coteˆ d’Azur, France Nikolaos Galatos Department of Mathematics, University of Denver, 2390 S. York St. Denver, CO 80210, USA George Metcalfe2,∗ Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland Simon Santschi Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland Abstract We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (ℓ-groups) if and only if it is valid in the variety DLM of distributive lattice- ordered monoids (distributive ℓ-monoids). This contrasts with the fact that, as proved by Repnitski˘ı, there exist inverse-free equations that are valid in all Abelian ℓ-groups but not in all commutative distributive ℓ-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations arXiv:2103.00146v1 [math.GR] 27 Feb 2021 in LG to the validity of equations in DLM. Keywords: Lattice-ordered groups, distributive lattice-ordered monoids, free ∗Corresponding author Email addresses: [email protected] (Almudena Colacito), [email protected] (Nikolaos Galatos), [email protected] (George Metcalfe), [email protected] (Simon Santschi) 1Supported by Swiss National Science Foundation (SNF) grant No. -
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. -
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 . -
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. -
Logic Machines and Diagrams
LOGIC M ACH IN ES AND DIAGRAMS Martin Gardner McGRAW-HILL BOOK COMPANY, INC. New York Toronto London 1958 LOGIC MACHINES AND DIAGRAMS Copyright 1958 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number: 58-6683 for C. G. Who thinks in a multivalued system all her own. Preface A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. The two fields are closely intertwined, and this book is the first attempt in any language to trace their curious, fascinating histories. Let no reader imagine that logic machines are merely the play- of who to have a recreational interest in things engineers happen , symbolic logic. As we move with terrifying speed into an age of automation, the engineers and mathematicians who design our automata constantly encounter problems that are less mathemati- cal in form than logical. It has been discovered, for example, that symbolic logic can be applied fruitfully to the design and simplification of switching circuits. It has been found that electronic calculators often require elaborate logic units to tell them what steps to follow in tackling certain problems. And in the new field of operations research, annoying situations are constantly arising for which techniques of symbolic logic are surprisingly appropriate. The last chaptej: of this book suggests some of the ways in which logic machines may play essential roles in coping with the stagger- ing complexities of an automated technology.