1st Edition

Universal Algebra Fundamentals and Selected Topics

By Clifford Bergman Copyright 2012
    320 Pages 56 B/W Illustrations
    by Chapman & Hall

    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.

    More about subs, homs, and prods
    Generating subalgebras
    Congruences and quotient algebras

    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
    Invariant relations
    Terms and free algebras
    Identities and Birkhoff’s theorem
    The lattice of subvarieties
    Equational theories and fully invariant congruences
    Maltsev conditions

    Congruence Distributive Varieties
    Ultrafilters and ultraproducts
    Jónsson’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
    Directly representable varieties revisited
    Minimal varieties
    Functionally complete algebras

    Finite Algebras and Locally Finite Varieties
    Minimal algebras
    Localization and induced algebras
    Centralizers again!




    Clifford Bergman is the Janson Professor of Mathematics at Iowa State University, where he has taught since 1982. He teaches both undergraduate and graduate courses in algebra and cryptography. Dr. Bergman’s research centers on classical questions in universal algebra, computational complexity, cryptology, and steganography.

    … as far as I am concerned, the book under review, by Clifford Bergman, is most welcome: we need more of this sort of thing, both for potential universal algebraists and for people like me: fellow travelers to some degree, or mathematicians who both use and thoroughly adore algebra and its structural qualities, and find themselves growing more appreciative of this architectural elegance as they evolve in their work and studies. … it is clearly written and pleasant to read … the author provides motivation as well as examples and exercises galore. At first glance it looks to me like the exercises are well-structured and should do the job of bringing the student or reader along at a decent pace from ignorance to both an appreciation for the subject and some facility with it. It’s definitely an area worth pursuing for a graduate student with the right disposition.
    —Michael Berg, MAA Reviews, December 2011

    … excellently written and is highly recommended to all who are interested in universal algebra.
    —Mathematical Reviews