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,
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,