A Course in Universal Algebra

Graduate Texts in Mathematics #78

by S. Burris, H P Sankappanavar

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.

Genres: Mathematics, Algebra

292 pages, Paperback

First published November 16, 1981

Reviews

[Rodrigo Almeida · Rating 5 out of 5]

A modern classic of the genre. The material is presented in an incredibly clear manner, getting the reader used to the level of abstraction (assuming a background of some algebra from undergraduate); it progresses to quite deep results, and research ideas - such as the hierarchy of complexity of varieties of universal algebras - which continue to shape the field today.
Am especially fond of the section on congruence distributive varieties, the one on Boolean products and the results on quasiprimal algebras, which certainly cannot be found in this manner in any other textbook.

[Lichen · Rating 5 out of 5]

One of my favourite books.