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Completely regular semigroups. (English) Zbl 0967.20034
Considered as a unary semigroup, a completely regular semigroup $$S$$ is a union of groups, where for every $$a\in S$$, $$a^{-1}$$ is the inverse of $$a$$ within the (uniquely determined) maximal subgroup of $$S$$ containing $$a$$. The completely regular semigroups form a variety $$\underline{CR}$$ determined by the associative law for the multiplication together with the identities $$xx^{-1}x=x$$, $$(x^{-1})^{-1}=x$$ and $$x^{-1}x=xx^{-1}$$. The variety of inverse semigroups which also satisfies the identities $$xx^{-1}x=x$$ and $$(x^{-1})^{-1}=x$$ has been the focus of much attention in the past four decades. While there is an abundance of natural examples of inverse semigroups, for completely regular semigroups the examples (beyond completely simple semigroups) are mostly artificially constructed: the minimum ideal of a finite semigroup is completely simple, and the various relatively free completely regular semigroups are the other more or less natural examples.
While for inverse semigroups there exists a well-developed structure theory tied in with a deep study of the lattice of congruences, the same cannot be said for completely regular semigroups. Useful constructions exist only for restricted classes of completely regular semigroups and the results concerning congruences arise as consequences from generalities which hold for regular semigroups in general. A large portion of this book is devoted to structure theory for such special classes of completely regular semigroups: completely simple semigroups, normal cryptogroups and regular orthogroups are investigated in Chapters 3, 4 and 5 respectively. The lattice of congruences of a completely regular semigroup is the topic of Chapters 6 and 7.
Clearly then, when dealing with completely regular semigroups, the universal algebraic aspects of the theory and possibly their intriguing connections with some group theoretical questions form the most attractive part of this area of investigation. The chapters which mainly deal with structure theory are interspersed with bits of information concerning parts of the lattice $$L(\underline{CR})$$ of completely regular semigroup varieties. Many of the results of Chapters 6 and 7 on the lattice of congruences are later applied to the study of the lattice of fully invariant congruences on a free completely regular semigroup and then give rise to various endomorphisms and complete congruences of the lattice $$L(\underline{CR})$$. Only two chapters are exclusively devoted to varieties of completely regular semigroups. Chapter 8 gives a fairly complete account of the present state of knowledge about the lattice of completely simple semigroup varieties, while Chapter 9 deals with Mal’cev products and concludes with a survey of several relevant endomorphisms of $$L(\underline{CR})$$.
The present work is the first volume in a planned two volume set and in the introduction the authors mention that the second volume will bring a deeper study of the lattice $$L(\underline{CR})$$. Missing so far is L. Polák’s description of $$L(\underline{CR})$$ as published in his trilogy of papers [Semigroup Forum 32, 97-123 (1985; Zbl 0564.20034); 36, No. 3, 253-284 (1987); 37, No. 1, 1-30 (1988; Zbl 0638.20032)] which by many is believed to be the centerpiece of the theory.

##### MSC:
 20M17 Regular semigroups 20M07 Varieties and pseudovarieties of semigroups 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20M19 Orthodox semigroups 20M05 Free semigroups, generators and relations, word problems 20M18 Inverse semigroups 08A30 Subalgebras, congruence relations 08B15 Lattices of varieties