PREFACE TO VOLUME II

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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