TABLE OF CONTENTS
PREFACE
IX
NOTATION USED IN VOLUME I
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CHAPTER 1. ELEMENTARY CONCEPTS
1.1 Basic definitions . . . . . . . . 1
1.2 Light's associativity test . . . . . . . 7
1.3 Translations and the regular representation . . . . 9
(Lemma 1.0-Theorem 1.3)
1.4 The semigroup of relations on a set . . . . . 1 3
(Lemma 1.4)
1.5 Congruences, factor groupoids and homomorphisms . . 16
(Theorem 1.5-Theorem 1.8)
1.6 Cyclic semigroups . . . . . . . . 1 9
(Theorem 1.9)
1.7 Units and maximal subgroups . . . . . 2 1
(Theorem 1.10-Theorem 1.11)
1.8 Bands and semilattices; bands of semigroups . . . 23
(Theorem 1.12)
1.9 Regular elements and inverses; inverse semigroups . . 26
(Lemma 1.13-Theorem 1.22)
1.10 Embedding semigroups in groups . . . . . 34
(Theorem 1.23-Theorem 1.25)
1.11 Right groups . . . . . . . . . 37
(Lemma 1.26-Theorem 1.27)
1.12 Free semigroups and generating relations; the bicyclic semi-
group . . . . . . . . . . 40
(Lemma 1.28-Corollary 1.32)
CHAPTER 2. IDEALS AND RELATED CONCEPTS
2.1 Green's relations . . . . . . . .
(Lemma 2.1-Theorem 2.4)
2.2 ^-structure of the full transformation semigroup &"x on a set X
(Lemma 2.5-Theorem 2.10)
2.3 Regulars-classes 58
(Theorem 2.11-Theorem 2.20)
47
51
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