Commutative semigroups. (English) Zbl 1040.20048

Grillet, P. A.

Advances in Mathematics (Dordrecht) 2. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-7067-8/hbk). xiv, 436 p. (2001).

The book is mainly devoted to developing the general structure theory of commutative semigroups. Several areas of the study of commutative semigroups are not covered: semigroup rings, factorization theory, ordered semigroups and varieties of commutative semigroups.
The author divides the book into three large parts. The first one contains the classical structure theory, which is arranged in eight chapters. The main chapters are 3, 4 and 5 which are devoted to semilattice decompositions, subdirect decompositions and group coextensions. This study is completed in chapter 6 (which contains Redei’s theorem), devoted to finitely generated semigroups.
The second part of the book has three chapters and deals with the author’s construction of subcomplete semigroups, including nilsemigroups and finitely generated semigroups, as a quotient of free semigroups by subcomplete congruences.
Finally, the last three chapters constitute the end part, containing cohomology theory for commutative semigroups.
An added value of the book is that it contains more than 800 references related to the subject.

Reviewer: Carlos Galindo (Castellon)

MSC:

20M14	Commutative semigroups
20-02	Research exposition (monographs, survey articles) pertaining to group theory

Keywords:

structure theory of commutative semigroups; semilattice decompositions; subdirect decompositions; group coextensions; cohomology theory for commutative semigroups