PREFACE

So far as we know, the term "semigroup" first appeared in mathematical

literature on page 8 of J.-A. de Siguier's book, filaments de la Theorie des

Groupes Abstraits (Paris, 1904), and the first paper about semigroups was a

brief one by L. E. Dickson in 1905. But the theory really began in 1928 with

the publication of a paper of fundamental importance by A. K. Susch-

kewitsch. In current terminology, he showed that every finite semigroup

contains a "kernel" (a simple ideal), and he completely determined the

structure of finite simple semigroups. A brief account of this paper is given

in Appendix A.

Unfortunately, this result of Suschkewitsch is not in a readily usable form.

This defect was removed by D. Rees in 1940 with the introduction of the

notion of a matrix over a group with zero, and, moreover, the domain of

validity was extended to infinite simple semigroups containing primitive

idempotents. The Rees Theorem is seen to be the analogue of Wedderburn's

Theorem on simple algebras, and it has had a dominating influence on the

later development of the theory of semigroups. Since 1940, the number of

papers appearing each year has grown fairly steadily to a little more than

thirty on the average.

It is in response to this developing interest that this book has been written.

Only one book has so far been published which deals predominantly with the

algebraic theory of semigroups, namely one by Suschkewitsch, The Theory

of Generalized Groups (Kharkow, 1937); this is in Russian, and is now out

of print. A chapter of R. H. Brack's A Survey of Binary Systems (Ergebnisse

der Math., Berlin, 1958) is devoted to semigroups. There is, of course, E.

Hille's book, Functional Analysis and Semi-groups (Amer. Math. Soc. Colloq.

PubL, 1948), and the 1957 revision thereof by Hille and R. S. Phillips;

but this deals with the analytic theory of semigroups and its application

to analysis. The time seems ripe for a systematic exposition of the

algebraic theory. (Since the above words were written, there has appeared

such an exposition, in Russian: Semigroups, by E. S. Lyapin, Moscow,

1960.)

The chief difficulty with such an exposition is that the literature is scat-

tered over extremely diverse topics. We have met this situation by con-

fining ourselves to a portion of the existing theory which has proved to be

capable of a well-knit and coherent development. All of Volume 1 and the

first half of Volume 2 center around the structure of semigroups of certain

types (such as simple semigroups, inverse semigroups, unions of groups,

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