Bergman Universal Algebra

Fundamentals and Selected Topics
1. Auflage 2011
ISBN: 978-1-4398-5130-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 320 Seiten

Reihe: Chapman & Hall Pure and Applied Mathematics

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 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.

Autoren/Hrsg.

Bergman, Clifford

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

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

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

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