Fundamentals of semigroup theory

John M. Howie0198511949, 9780198511946

This book is an indispensable source for anyone with an interest in semigroup theory or whose research overlaps with this increasingly important and active field of mathematics. It clearly emphasizes “pure” semigroup theory, in particular the various classes of regular semigroups. More than 150 exercises, accompanied by relevant references to the literature, give pointers to areas of the subject not explicitly covered in the text.

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Table of contents :
Title page……Page 1
Series……Page 2
Date-line……Page 4
Dedication……Page 5
Preface……Page 7
Contents……Page 9
1.1 Basic definitions……Page 11
1.2 Monogenic semigroups……Page 18
1.3 Ordered sets, semilattices and lattices……Page 23
1.4 Binary relations; equivalences……Page 26
1.5 Congruences……Page 32
1.6 Free semigroups and monoids; presentations……Page 39
1.7 Ideals and Rees congruences……Page 43
1.8 Lattices of equivalences and congruences*……Page 44
1.9 Exercises……Page 47
1.10 Notes……Page 54
2.1 Green’s equivalences……Page 55
2.2 The structure of $mathcal{D}$-classes……Page 58
2.3 Regular $mathcal{D}$-classes……Page 60
2.4 Regular semigroups……Page 64
2.5 The sandwich set……Page 68
2.6 Exercises……Page 70
2.7 Notes……Page 74
3.1 Simple and 0-simple semigroups; principal factors……Page 76
3.2 The Rees Theorem……Page 79
3.3 Completely simple semigroups……Page 87
3.4 Isomorphism and normalization……Page 90
3.5 Congruences on completely 0-simple semigroups*……Page 93
3.6 The lattice of congruences on a completely 0-simple semigroup*……Page 101
3.7 Finite congruence-free semigroups*……Page 103
3.8 Exercises……Page 105
3.9 Notes……Page 111
4 Completely regular semigroups……Page 112
4.1 The Clifford decomposition……Page 113
4.2 Clifford semigroups……Page 117
4.3 Varieties……Page 118
4.4 Bands……Page 123
4.5 Free bands……Page 129
4.6 Varieties of bands*……Page 134
4.7 Exercises……Page 148
4.8 Notes……Page 152
5 Inverse semigroups……Page 154
5.1 Preliminaries……Page 155
5.2 The natural order relation……Page 162
5.3 Congruences on inverse semigroups……Page 164
5.4 The Munn semigroup……Page 172
5.5 Anti-uniform semilattices……Page 176
5.6 Bisimple inverse semigroups……Page 179
5.7 Simple inverse semigroups……Page 186
5.8 Representations of inverse semigroups……Page 195
5.9 $E$-unitary inverse semigroups……Page 202
5.10 Free inverse monoids……Page 210
5.11 Exercises……Page 221
5.12 Notes……Page 229
6 Other classes of regular semigroups……Page 232
6.1 Locally inverse semigroups……Page 233
6.2 Orthodox semigroups……Page 236
6.3 Semibands……Page 240
6.4 Exercises……Page 245
6.5 Notes……Page 247
7.1 Properties of free semigroups……Page 248
7.2 Codes……Page 253
7.3 Exercises……Page 258
7.4 Notes……Page 260
8 Semigroup amalgams……Page 261
8.1 Systems……Page 262
8.2 Free products……Page 268
8.3 Dominions and zigzags……Page 276
8.4 Direct limits, free extensions and free products……Page 284
8.5 The extension property……Page 298
8.6 Inverse semigroups and amalgamation……Page 313
8.7 Exercises……Page 320
8.8 Notes……Page 325
References……Page 328
List of symbols……Page 351
Index……Page 355