Universal algebra : fundamentals and selected topics
- Bergman, C. H. (Clifford H.), 1953-
- Boca Raton, FL : CRC Press, c2012.
- Physical description
- xi, 308 p. : ill ; 25 cm.
- Monographs and textbooks in pure and applied mathematics ; 301.
QA251 .B45 2012
- Unknown QA251 .B45 2012
- Includes bibliographical references (p. 291-297) and indexes.
- FUNDAMENTALS OF UNIVERSAL ALGEBRA Algebras Operations Examples More about subs, homs, and prods Generating subalgebras Congruences and quotient algebras Lattices Ordered sets Distributive and modular lattices Complete lattices Closure operators and algebraic lattices Galois connections Ideals in lattices The Nuts and Bolts of Universal Algebra The isomorphism theorems Direct products Subdirect products Case studies Varieties and other classes of algebras Clones, Terms, and Equational Classes Clones Invariant relations Terms and free algebras Identities and Birkhoff's theorem The lattice of subvarieties Equational theories and fully invariant congruences Maltsev conditions Interpretations SELECTED TOPICS Congruence Distributive Varieties Ultrafilters and ultraproducts Jonsson's lemma Model theory Finitely based and nonfinitely based algebras Definable principal (sub)congruences Arithmetical Varieties Large clones How rare are primal algebras? Maltsev Varieties Directly representable varieties The centralizer congruence Abelian varieties Commutators Directly representable varieties revisited Minimal varieties Functionally complete algebras Finite Algebras and Locally Finite Varieties Minimal algebras Localization and induced algebras Centralizers again! Applications Bibliography Index.
- (source: Nielsen Book Data)
- Publisher's Summary
- 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 Jonsson'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 Murskii'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.
(source: Nielsen Book Data)
- Publication date
- Clifford Bergman.
- Pure and applied mathematics
- Monographs and textbooks in pure and applied mathematics ; .301
- "A Chapman & Hall book."