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Semigroups with a system of subsemigroups. (English) Zbl 0569.20049
Novi Sad, Yugoslavia: University of Novi Sad, Institute of Mathematics. V, 196 p. (1985).
Every chapter of the book - except for the first two and the VIIth which are of introductory character - deals with some particular class of semigroups. The author’s intention was to exclude topics exposed in one of the monographs in common use. The main scope of his interest is giving different characterizations of the considered classes. Sometimes (unfortunately not very often) this characterization is given by a structure theorem. The following classes are dealt with (along with some of their subclasses): semigroups whose proper ideals are Archimedean (Ch. III); \(\pi\)-regular and \(\pi\)-inverse semigroups, i.e. such that some power of every element is regular (Ch. IV-V) and, in particular, those of them every regular element of which is completely regular (called GV- semigroups: they turn out to coincide with semilattices of completely Archimedean semigroups, Ch. X); nil-extensions of simple semigroups (Ch. VI); semigroups in which every proper left ideal is completely simple and those in which every subsemigroup is an (m,n)-ideal (Ch. VIII); bands of (right) Archimedean and of power joined semigroups (Ch. IX); semigroups in which some power of each element lies in a dihedral group (Ch. XI). An Appendix describes an algorithm for obtaining all semigroups of order n, and lists the semigroups of order \(n\leq 5\) (up to antiisomorphism). There is a number of exercises at the ends of the paragraphs.
Reviewer: G.Pollák

20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
20-02 Research exposition (monographs, survey articles) pertaining to group theory