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Author: S. Burris
Publisher: Springer
ISBN: 9781461381327
Category : Mathematics
Languages : en
Pages : 276
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Book Description
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.
Author: S. Burris
Publisher: Springer
ISBN: 9781461381327
Category : Mathematics
Languages : en
Pages : 276
Get Book
Book Description
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.
Author: George M. Bergman
Publisher: Springer
ISBN: 3319114786
Category : Mathematics
Languages : en
Pages : 574
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Book Description
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.
Author: Clifford Bergman
Publisher: CRC Press
ISBN: 1439851298
Category : Computers
Languages : en
Pages : 324
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Book Description
Starting with the most basic notions, Universal Algebra: Fundamentals and Selected Topics introduces all the key elements needed to read and understand current research in this field. Based on the author’s two-semester course, the text prepares students for research work by providing a solid grounding in the fundamental constructions and concepts of universal algebra and by introducing a variety of recent research topics. The first part of the book focuses on core components, including subalgebras, congruences, lattices, direct and subdirect products, isomorphism theorems, a clone of operations, terms, free algebras, Birkhoff’s theorem, and standard Maltsev conditions. The second part covers topics that demonstrate the power and breadth of the subject. The author discusses the consequences of Jónsson’s lemma, finitely and nonfinitely based algebras, definable principal congruences, and the work of Foster and Pixley on primal and quasiprimal algebras. He also includes a proof of Murskiĭ’s theorem on primal algebras and presents McKenzie’s characterization of directly representable varieties, which clearly shows the power of the universal algebraic toolbox. The last chapter covers the rudiments of tame congruence theory. Throughout the text, a series of examples illustrates concepts as they are introduced and helps students understand how universal algebra sheds light on topics they have already studied, such as Abelian groups and commutative rings. Suitable for newcomers to the field, the book also includes carefully selected exercises that reinforce the concepts and push students to a deeper understanding of the theorems and techniques.
Author: Dietlinde Lau
Publisher: Springer Science & Business Media
ISBN: 3540360239
Category : Mathematics
Languages : en
Pages : 668
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Book Description
Function Algebras on Finite Sets gives a broad introduction to the subject, leading up to the cutting edge of research. The general concepts of the Universal Algebra are given in the first part of the book, to familiarize the reader from the very beginning on with the algebraic side of function algebras. The second part covers the following topics: Galois-connection between function algebras and relation algebras, completeness criterions, and clone theory.
Author: Ralph N. McKenzie
Publisher: American Mathematical Society
ISBN: 1470442957
Category : Mathematics
Languages : en
Pages : 386
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Book Description
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.
Author: Jorge Almeida
Publisher: World Scientific
ISBN: 9814501565
Category : Mathematics
Languages : en
Pages : 532
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Book Description
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. Contents:Finite Universal Algebra:Elements of Universal AlgebraOrder and TopologyFinite AlgebrasDecidabilityFinite Semigroups and Monoids:PreliminariesPermutativityOperators Relating Semigroups and MonoidsSemigroups Whose Regular D-Classes are SubsemigroupsThe JoinThe Semidirect ProductThe PowerFactorization of Implicit OperationsOpen Problems Readership: Mathematicians and computer scientists. keywords:Inite Semigroups;Finite Monoids;Universal Algebra;Recognizable Languages;Pseudovarieties;Pseudoidentities;Implicit Operations;Relatively Free Profinite Semigroups;Semidirect Products;Power Semigroups “This book is devoted to an exciting new field where author has made important contributions, and thus it is a most welcome addition to the existing literature. It will find its place on the bookshelves of many a specialist in semigroups, as well as species of algebraists and computer scientists, including graduate students.” Semigroup Forum “The book … constitutes an important contribution to the most active part of the present theory of finite semigroups. All overwhelming majority of the results included in it is very new and has been scattered over journals so far. The book does not cover all of the theory of semigroup … but it is extremely rich in material and ideas presented with skill and dedication. The book has already influenced the area essentially, and its influence will certainly grow … I think the book is a must for researchers in the area but it is also very useful for all those who want to trace modern developments in the theory of semigroups.” Mathematics Abstracts
Author: Stanley Burris
Publisher:
ISBN: 9783540905783
Category : Algebra, Universal
Languages : en
Pages : 276
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Book Description
Author: Richard S Pierce
Publisher: Courier Corporation
ISBN: 0486789985
Category : Mathematics
Languages : en
Pages : 162
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Book Description
"Suitable for introductory graduate-level courses and independent study, this text presents the basic definitions of the theory of abstract algebra. Following introductory material, each of four chapters focuses on a major theme of universal algebra: subdirect decompositions, direct decompositions, free algebras, and varieties of algebra. Problems and a bibliography supplement the text. "--
Author: Stanley Burris
Publisher:
ISBN: 9780988055209
Category :
Languages : en
Pages : 276
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Book Description
Author: J. P. May
Publisher: University of Chicago Press
ISBN: 9780226511832
Category : Mathematics
Languages : en
Pages : 262
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Book Description
Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.
