Lattices, Semigroups, and Universal Algebra
Lattices, Semigroups, and Universal Algebra

Author: Jorge Almeida

Publisher: Springer Science & Business Media

Published: 2013-11-11

Total Pages: 325

ISBN-13: 1489926089

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This volume contains papers which, for the most part, are based on talks given at an international conference on Lattices, Semigroups, and Universal Algebra that was held in Lisbon, Portugal during the week of June 20-24, 1988. The conference was dedicated to the memory of Professor Antonio Almeida Costa, a Portuguese mathematician who greatly contributed to the development of th algebra in Portugal, on the 10 anniversary of his death. The themes of the conference reflect some of his research interests and those of his students. The purpose of the conference was to gather leading experts in Lattices, Semigroups, and Universal Algebra and to promote a discussion of recent developments and trends in these areas. All three fields have grown rapidly during the last few decades with varying degrees of interaction. Lattice theory and Universal Algebra have historically evolved alongside with a large overlap between the groups of researchers in the two fields. More recently, techniques and ideas of these theories have been used extensively in the theory of semigroups. Conversely, some developments in that area may inspire further developments in Universal Algebra. On the other hand, techniques of semi group theory have naturally been employed in the study of semilattices. Several papers in this volume elaborate on these interactions.

A Course in Universal Algebra
A Course in Universal Algebra

Author: S. Burris

Publisher: Springer

Published: 2011-10-21

Total Pages: 276

ISBN-13: 9781461381327

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Universal algebra has enjoyed a particularly explosive growth in the last twenty years, and a student entering the subject now will find a bewildering amount of material to digest. This text is not intended to be encyclopedic; rather, a few themes central to universal algebra have been developed sufficiently to bring the reader to the brink of current research. The choice of topics most certainly reflects the authors' interests. Chapter I contains a brief but substantial introduction to lattices, and to the close connection between complete lattices and closure operators. In particular, everything necessary for the subsequent study of congruence lattices is included. Chapter II develops the most general and fundamental notions of uni versal algebra-these include the results that apply to all types of algebras, such as the homomorphism and isomorphism theorems. Free algebras are discussed in great detail-we use them to derive the existence of simple algebras, the rules of equational logic, and the important Mal'cev conditions. We introduce the notion of classifying a variety by properties of (the lattices of) congruences on members of the variety. Also, the center of an algebra is defined and used to characterize modules (up to polynomial equivalence). In Chapter III we show how neatly two famous results-the refutation of Euler's conjecture on orthogonal Latin squares and Kleene's character ization of languages accepted by finite automata-can be presented using universal algebra. We predict that such "applied universal algebra" will become much more prominent.

M-Solid Varieties of Algebras
M-Solid Varieties of Algebras

Author: Jörg Koppitz

Publisher: Springer Science & Business Media

Published: 2006-02-10

Total Pages: 364

ISBN-13: 9780387308043

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A complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on solid varieties of semirings and semigroups. The book aims to develop the theory of solid varieties as a system of mathematical discourse that is applicable in several concrete situations. A unique feature of this book is the use of Galois connections to integrate different topics.

Algebras, Lattices, Varieties
Algebras, Lattices, Varieties

Author: Ralph N. McKenzie

Publisher: American Mathematical Society

Published: 2018-07-09

Total Pages: 386

ISBN-13: 1470442957

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This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text.

An Invitation to General Algebra and Universal Constructions
An Invitation to General Algebra and Universal Constructions

Author: George M. Bergman

Publisher: Springer

Published: 2015-02-05

Total Pages: 574

ISBN-13: 3319114786

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Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.

Finite Semigroups And Universal Algebra
Finite Semigroups And Universal Algebra

Author: Jorge Almeida

Publisher: World Scientific

Published: 1995-01-27

Total Pages: 532

ISBN-13: 9814501565

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Motivated by applications in theoretical computer science, the theory of finite semigroups has emerged in recent years as an autonomous area of mathematics. It fruitfully combines methods, ideas and constructions from algebra, combinatorics, logic and topology. In simple terms, the theory aims at a classification of finite semigroups in certain classes called “pseudovarieties”. The classifying characteristics have both structural and syntactical aspects, the general connection between them being part of universal algebra. Besides providing a foundational study of the theory in the setting of arbitrary abstract finite algebras, this book stresses the syntactical approach to finite semigroups. This involves studying (relatively) free and profinite free semigroups and their presentations. The techniques used are illustrated in a systematic study of various operators on pseudovarieties of semigroups.

Universal Algebra
Universal Algebra

Author: George Grätzer

Publisher: Springer Science & Business Media

Published: 2008-12-15

Total Pages: 601

ISBN-13: 0387774874

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Universal Algebra has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well selected additional bibliography of over 1250 papers and books which makes this an indispensable new edition for students, faculty, and workers in the field.

The Shape of Congruence Lattices
The Shape of Congruence Lattices

Author: Keith Kearnes

Publisher: American Mathematical Soc.

Published: 2013-02-26

Total Pages: 183

ISBN-13: 0821883232

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This monograph is concerned with the relationships between Maltsev conditions, commutator theories and the shapes of congruence lattices in varieties of algebras. The authors develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. They prove that a residually small variety that satisfies a congruence identity is congruence modular.