Classical Finite Transformation Semigroups (eBook)

Preface 6
Contents 9
Ordinary and Partial Transformations 13
1.1 Basic Definitions 13
1.2 Graph of a (Partial) Transformation 15
1.3 Linear Notation for Partial Transformations 20
1.4 Addenda and Comments 22
1.5 Additional Exercises 25
The Semigroups Tn, PT n,and ISn 27
2.1 Composition of Transformations 27
2.2 Identity Elements 29
2.3 Zero Elements 31
2.4 Isomorphism of Semigroups 32
2.5 The Semigroup 34
2.6 Regular and Inverse Elements 35
2.7 Idempotents 38
2.8 Nilpotent Elements 40
2.9 Addenda and Comments 43
2.10 Additional Exercises 47
Generating Systems 51
3.1 Generating Systems in T n, PT n, and ISn 51
3.2 Addenda and Comments 54
3.3 Additional Exercises 55
Ideals and Green’s Relations 56
4.1 Ideals of Semigroups 56
4.2 Principal Ideals in T n, PT n, and ISn 57
4.3 Arbitrary Ideals in T n, PT n, and ISn 61
4.4 Green’s Relations 64
4.5 Green’s Relations on T n, PT n, and ISn 69
4.6 Combinatorics of Green’s Relationsin the Semigroups T n, PT n, and ISn 71
4.7 Addenda and Comments 73
4.8 Additional Exercises 76
Subgroups and Subsemigroups 79
5.1 Subgroups 79
5.2 Cyclic Subsemigroups 80
5.3 Isolated and Completely Isolated Subsemigroups 84
5.4 Addenda and Comments 93
5.5 Additional Exercises 98
Other Relations on Semigroups 100
6.1 Congruences and Homomorphisms 100
6.2 Congruences on Groups 103
6.3 Congruences on T n, PT n, and ISn 105
6.4 Conjugate Elements 112
6.5 Addenda and Comments 117
6.6 Additional Exercises 119
Endomorphisms 120
7.1 Automorphisms of T n, PT n, and ISn 120
7.2 Endomorphisms of Small Ranks 123
7.3 Exceptional Endomorphism 124
7.4 Classification of Endomorphisms 127
7.5 Combinatorics of Endomorphisms 132
7.6 Addenda and Comments 136
7.7 Additional Exercises 137
Nilpotent Subsemigroups 139
8.1 Nilpotent Subsemigroups and Partial Orders 139
8.2 Classification of Maximal Nilpotent Subsemigroups 142
8.3 Cardinalities of Maximal Nilpotent, Subsemigroups 146
8.4 Combinatorics of Nilpotent Elements in ISn 149
8.5 Addenda and Comments 156
8.6 Additional Exercises 159
Presentation 161
9.1 Defining Relations 161
9.2 A presentation for ISn 164
9.3 A Presentation for T n 169
9.4 A presentation for PT n 177
9.5 Addenda and Comments 180
9.6 Additional Exercises 181
Transitive Actions 183
10.1 Action of a Semigroup on a Set 183
10.2 Transitive Actions of Groups 185
10.3 Transitive Actions of T n 187
10.4 Actions Associated with L-Classes 188
10.5 Transitive Actions of PT n and ISn 190
10.6 Addenda and Comments 193
10.7 Additional Exercises 195
Linear Representations 197
11.1 Representations and Modules 197
11.2 L-Induced S-Modules 200
11.3 Simple Modules over ISn and PT n 203
11.4 Effective Representations 206
11.5 Arbitrary ISn-Modules 208
11.6 Addenda and Comments 212
11.7 Additional Exercises 219
Cross-Sections 222
12.1 Cross-Sections 222
12.2 Retracts 223
12.3 H-Cross-Sections in T n, PT n, and ISn 226
12.4 L-Cross-Sections in T n and PT n 229
12.5 L-Cross-Sections in ISn 233
12.6 R-Cross-Sections in ISn 238
12.7 Addenda and Comments 240
12.8 Additional Exercises 242
Variants 244
13.1 Variants of Semigroups 244
13.2 Classification of Variants for ISn, T n,and PT n 247
13.3 Idempotents and Maximal Subgroups 250
13.4 Principal Ideals and Green’s Relations 252
13.5 Addenda and Comments 253
13.6 Additional Exercises 256
Order-Related Subsemigroups 258
14.1 Subsemigroups, Related to the Natural Order 258
14.2 Cardinalities 260
14.3 Idempotents 264
14.4 Generating Systems 267
14.5 Addenda and Comments 274
14.6 Additional Exercises 280
Answers and Hints to Exercises 283
Bibliography 289
List of Notation 302
Index 311