In broad outline, Volume 2 follows the plan predicted in Volume 1.
Cross-references to Chapters 6, 7, and 8, from Volume 1, remain correct. The
original Chapter 9 has expanded into the present Chapters 9, 10, and 11, and
references from Volume 1 should be interpreted accordingly. Chapter 12 is
the new name of the chapter referred to as Chapter 10 in Volume 1.
Volumes 1 and 2 should be thought of as a single work presenting a survey
of the theory of semigroups. The greater part of Volume 2 deals not with the
deeper development of the topics initiated in Volume 1, but with additional
branches of the theory to which there was at most passing reference in
Volume 1. Most of the subject matter of Volume 2 is taken from papers
published prior to the drawing up of the original plan for both volumes.
Nevertheless, the chance has been taken, on the topics with which Volume 2
deals, to include what we judge to be the more important developments to
date. On the other hand, the theory of matrix representations of semigroups
(Chapter 5) has seen important extensions since Volume 1 appeared (see, in
particular, Munn [1964b] and the references therein); but we have not pre-
sented these in the present volume.
Among the more important recent developments of which we present an
extended treatment are B. M. Sam's theory of the representations of an
arbitrary semigroup by partial one-to-one transformations of a set (§§7.2, 7.3,
11.4), L. Redei's theory of finitely generated commutative semigroups, to
which we give an introduction (§9.2) (our account is based on a lecture de-
livered by Professor Redei at Oxford in 1960; unfortunately we were unable
to obtain a copy of Redei's book, Theorie der endlich erzeugbaren kommutativen
Halbgruppen [1963], before our manuscript went to the printer), J. M. Howie's
theory of amalgamated free products of semigroups (§9.4), and E. J. Tully's
theory of representations of a semigroup by transformations of a set
(Chapter 11).
In §10.8 we present Malcev's [1952] theory of the congruences on a full
transformation semigroup and in §§12.6 and 12.8 his [1937, 1939, 1940] dis-
cussion of necessary and sufficient conditions for the embeddability of a semi-
group in a group. In both cases our account follows in general plan the argu-
ment of Malcev's brilliant papers; but we feel that the considerable amplifica-
tion our account contains is necessary for complete proofs (which we hope do
not now require further amplification).
The material in § 7.4, Chapter 9, §§10.7 and 10.8 was presented by one of
the authors in a course of lectures, under the title Congruences on Semigroups,
to the National Science Foundation Summer Institute in Algebra, June 24-
August 16, 1963, at Pennsylvania State University. An account of these
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