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Book review

Author(s) Rosenberg, Ivo G. (ed.)
Sabidussi, Gert (ed.)
Title Algebras and Orders - Proceedings of the NATO Advanced Study Institute and Séminaire de mathématiques supérieures on Algebras and Orders, Montreal, Canada, July 29 - August 9, 1991
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by Jan Paseka

The birth of abstract algebra can be traced to a paper of Cayley on group theory (1854). Universal algebra established itself in the 1930's as a unifying tool for the theory of groups, modules, rings and lattices. At this time the logicians A. Lindenbaum and A. Tarski proved the representation theorem for complete atomic Boolean algebras. The topologist M. Stone introduced the notion of topology to algebra and proved his famous Stone's representation theorem for Boolean algebras. The algebraist G. Birkhoff proved his Preservation Theorem (1935) which asserts that every class of algebras is closed under the operations of forming homomorphic images, subalgebras and direct products is an equational class (variety).

The purpose of this book is to present introductory and surveying articles on topics covering the links of universal algebra to boolean algebra, lattice theory, topology, graphs, relations, automata, theoretical computer science and orders. The articles are accessible to everyone who is or will be interested in some of presented topics. Let us list the 11 articles and their authors: Peter Burmeister, Partial algebras - an introductory survey, Brian A. Davey, Duality theory on ten dollars a day, Marcel Erné, Algebraic ordered sets and their generalizations, Isidore Fleischer, A Boolean formalization of predicate calculus, Ralph Freese, Lectures on free lattices, Bjarni Jónsson, A survey of Boolean algebras with operators, H. Machida, Ivo G. Rosenberg, Essentially minimal groupoids, Alden F. Pixley, Functional and affine completeness and arithmetical varieties, Ivan Rival, Reading, drawing, and order, Dietmar Schweigert, Hyperidentities, Walter Taylor, Abstract clone theory.

The level of all articles is high - they are addressed to graduates which would like become more acquainted with the directions listed in the articles. All authors are experts in the presented field.

This is a valuable book which most departments of mathematics should have in their library. The book was printed on acid-free paper and its typesetting quality is high - all articles are written in TeX. The index of the book has seven pages. All in all purchasing this book is a good investment for everyone interested in universal algebra and/or ordered sets.