×

Semigroups. An introduction to the structure theory. (English) Zbl 0830.20079

Pure and Applied Mathematics, Marcel Dekker. 193. New York, NY: Marcel Dekker, Inc. ix, 398 p. $ 150.00 (1995).
The definitive work on the theory of semigroups, one might say “the Bible”, is the book of A. H. Clifford and G. B. Preston [The algebraic theory of semigroups. Volumes I and II. (1961; Zbl 0111.03403 and 1967; Zbl 0178.01203)]. The very clever idea of the present author was to write a book on the development of the theory of semigroups after the publication of “the Bible”.
The present book is devoted to constructions and descriptions of semigroups, with emphasis on the four classes for which the structure theory has been most successful: finite semigroups, commutative semigroups, regular semigroups, and inverse semigroups.
The first three chapters contain basic results; the other six chapters are largely independent of each other. The latter chapter headings are as follows: Commutative semigroups, Finite semigroups, Regular semigroups, Inverse semigroups, Fundamental regular semigroups, Four classes of regular semigroups.
Reviewer’s remarks:
1) In the present book the number of references is ca. half as many as in the two volumes of Clifford and Preston. The papers of authors who were quite active in the last thirty years in the field of semigroups (e.g. L. M. Gluskin, Š. Schwarz, V. V. Vagner) were completely ignored by the present author.
2) The notion of quasi- ideal introduced by O. Steinfeld and his book [Quasi-ideals in rings and semigroups. Budapest: Akadémiai Kiadó, (1978; Zbl 0403.16001)] are not mentioned in the text and in the Bibliography.
3) K. H. Kim wrote several papers on the semigroup of binary relations. Neither his activities nor his book [Boolean matrix theory and applications. Marcel Dekker (1982; Zbl 0495.15003)] are mentioned in the book under review. It is funny since Kim’s book has been published by the same publisher and in the same series as the book under review.
4) Among generalized inverses the so called Drazin inverse plays an important role. It was introduced by M. P. Drazin [in his paper Am. Math. Mon. 65, 506-514 (1958; Zbl 0083.02901)]. Later it was investigated by several authors. The reader of this book will not be able to learn about Drazin inverses or the recent results in this field from this book. After that it is not surprising that the generalized inverse introduced by the reviewer is also not mentioned. The reviewer’s inverse is called by him quasiinverse, it is very similar to the Drazin inverse, but not as widely known as the Drazin inverse. The interested reader, who might be interested in quasiinverses, may read about them in the reviewer’s papers [see Theory Graphs, Sympos. Rome 1966, 93–101 (1967; Zbl 0188.05701), Graph Theory, Proc. Colloq. Tihany, Hungary 1966, 65–75 (1968; Zbl 0159.02801), Publ. Math. 15, 283–285 (1969; Zbl 0175.01701)].

MSC:

20Mxx Semigroups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20M10 General structure theory for semigroups
20M17 Regular semigroups
20M18 Inverse semigroups
20M14 Commutative semigroups