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