##
**Inverse semigroups.**
*(English)*
Zbl 0546.20053

Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. X, 674 p. £63.95 (1984).

This is the first book on the theory of inverse semigroups, a theory which appeared in its abstract setting in 1952-54, and which has been considered in various disguises from at least the thirties. The size, price, thouroughness of proofs, abundance of material (distributed into 14 chapters) - all that impresses the reader. As the author mentions in the preface, ”[t]he choice of the material and the form of its presentation is primarily geared to exhibit the method of investigation rather than to achieve the greatest possible generality in the shortest possible way. This entails treating some subjects in more detail than they seem to deserve from the point of view of their intrinsic value.”

Chapter I contains some preliminary results of the theory of semigroups, and in Chapter II definitions and elementary properties from the theory of inverse semigroups are given together with a description of four special classes of semigroups: Clifford semigroups, Brandt semigroups, strict inverse semigroups \([=subdirect\) products of Brandt inverse semigroups and groups], and Reilly semigroups \([=bisimple \omega\) - semigroups].

Chapter III is dedicated to congruences. Every congruence can be uniquely determined by the set of those congruence classes which contain idempotents or by the set of all elements congruent to idempotents together with the congruence on the semilattice of idempotents induced by the original congruence. The lattice of all congruences and several types of special congruences are considered.

The historic motivation for introduction of the class of inverse semigroups and preliminary results of the theory obtained before 1952 were connected with inverse semigroups of one-to-one partial transformations. In Chapter IV the author considers (homomorphic and isomorphic) representations of inverse semigroups by such transformations.

In Chapters V and VI various ”hulls” of an inverse semigroup are considered: the translational hull (introduced for general semigroups by Gluskin who developed earlier results of Lyapin), and conjugate and normal hulls which, in the case of general inverse semigroups, were introduced by the author of the book.

Chapter VII is dedicated to the McAlister P-theorem and related results. [The present reviewer cannot restrain himself from illustrating the author’s brief introductory remark on some difference in ”Weltanschauungen” between the Soviet and Western schools in the theory of semigroups: neither in this book nor in any other Western sources has the reviewer seen references to the ”pre-Wagner history” of the theory of inverse semigroups. Enough is to say that a very revealing discussion leading directly to the P-theorem was published in a widely circulated Western journal in... 1939. It is arguable whether this oblivion of the sources and powerful driving forces which appeared in other parts of mathematics and naturally led to the theory of inverse semigroups serves the best interests of this part of algebra.]

Chapters VIII-XI treat other special classes of inverse semigroups: free (in Chapter VIII), monogenic (in Chapter IX), bisimple inverse monoids (in Chapter X), and \(\omega\) -regular (in Chapter XI).

Theory of varieties of inverse semigroups is in its initial steps. Some of the known results (which were fairly complete by the time the book was written) are presented in Chapter XII. Chapter XIII is dedicated to amalgamation of inverse semigroups, and Chapter XIV presents a construction of inverse semigroups, from partial groupoids obtained by restricting the multiplication of an inverse semigroup to a partial operation. [Another historic remark may be in place here. It illustrates the influence of your semigroup Weltanschauung. The construction of inverse semigroups in terms of partial groupoids was something almost self-obvious for the Soviet school of inverse semigroups which grew on the foundation of differential geometry. The reviewer remembers quite vividly that he discussed this construction in a paper written in 1962 and published in 1965 only because it was used to solve another problem. By itself, it hardly seemed to deserve mention, and it was buried in a Russian provincial publication for the same reason. However, the situation changes drastically once you are cut off from the sources of the theory of inverse semigroups. The construction ceases to be obvious. It has been rediscovered independently in the West in 1976. The irony of the situation lies in the fact that the sources of the theory of inverse semigroups are an entirely Western creation which has been transplanted to the Russian soil in the late forties. The time may be right for somebody to write a paper recalling all that. A known adage says that new things are, in fact, the old ones which have been very well forgotten.]

The book contains a great number of exercises which complement the main exposition. There are quite a few open problems, and the bibliography contains more than 500 items.

Every semigroup-theorist (and many people in other fields) would like very much to have a copy of this book on their shelves.

Chapter I contains some preliminary results of the theory of semigroups, and in Chapter II definitions and elementary properties from the theory of inverse semigroups are given together with a description of four special classes of semigroups: Clifford semigroups, Brandt semigroups, strict inverse semigroups \([=subdirect\) products of Brandt inverse semigroups and groups], and Reilly semigroups \([=bisimple \omega\) - semigroups].

Chapter III is dedicated to congruences. Every congruence can be uniquely determined by the set of those congruence classes which contain idempotents or by the set of all elements congruent to idempotents together with the congruence on the semilattice of idempotents induced by the original congruence. The lattice of all congruences and several types of special congruences are considered.

The historic motivation for introduction of the class of inverse semigroups and preliminary results of the theory obtained before 1952 were connected with inverse semigroups of one-to-one partial transformations. In Chapter IV the author considers (homomorphic and isomorphic) representations of inverse semigroups by such transformations.

In Chapters V and VI various ”hulls” of an inverse semigroup are considered: the translational hull (introduced for general semigroups by Gluskin who developed earlier results of Lyapin), and conjugate and normal hulls which, in the case of general inverse semigroups, were introduced by the author of the book.

Chapter VII is dedicated to the McAlister P-theorem and related results. [The present reviewer cannot restrain himself from illustrating the author’s brief introductory remark on some difference in ”Weltanschauungen” between the Soviet and Western schools in the theory of semigroups: neither in this book nor in any other Western sources has the reviewer seen references to the ”pre-Wagner history” of the theory of inverse semigroups. Enough is to say that a very revealing discussion leading directly to the P-theorem was published in a widely circulated Western journal in... 1939. It is arguable whether this oblivion of the sources and powerful driving forces which appeared in other parts of mathematics and naturally led to the theory of inverse semigroups serves the best interests of this part of algebra.]

Chapters VIII-XI treat other special classes of inverse semigroups: free (in Chapter VIII), monogenic (in Chapter IX), bisimple inverse monoids (in Chapter X), and \(\omega\) -regular (in Chapter XI).

Theory of varieties of inverse semigroups is in its initial steps. Some of the known results (which were fairly complete by the time the book was written) are presented in Chapter XII. Chapter XIII is dedicated to amalgamation of inverse semigroups, and Chapter XIV presents a construction of inverse semigroups, from partial groupoids obtained by restricting the multiplication of an inverse semigroup to a partial operation. [Another historic remark may be in place here. It illustrates the influence of your semigroup Weltanschauung. The construction of inverse semigroups in terms of partial groupoids was something almost self-obvious for the Soviet school of inverse semigroups which grew on the foundation of differential geometry. The reviewer remembers quite vividly that he discussed this construction in a paper written in 1962 and published in 1965 only because it was used to solve another problem. By itself, it hardly seemed to deserve mention, and it was buried in a Russian provincial publication for the same reason. However, the situation changes drastically once you are cut off from the sources of the theory of inverse semigroups. The construction ceases to be obvious. It has been rediscovered independently in the West in 1976. The irony of the situation lies in the fact that the sources of the theory of inverse semigroups are an entirely Western creation which has been transplanted to the Russian soil in the late forties. The time may be right for somebody to write a paper recalling all that. A known adage says that new things are, in fact, the old ones which have been very well forgotten.]

The book contains a great number of exercises which complement the main exposition. There are quite a few open problems, and the bibliography contains more than 500 items.

Every semigroup-theorist (and many people in other fields) would like very much to have a copy of this book on their shelves.

Reviewer: B.M.Schein

### MSC:

20M10 | General structure theory for semigroups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20M20 | Semigroups of transformations, relations, partitions, etc. |

20M15 | Mappings of semigroups |

20M07 | Varieties and pseudovarieties of semigroups |