Ganyushkin / Mazorchuk Classical Finite Transformation Semigroups

An Introduction
1. Auflage 2008
ISBN: 978-1-84800-281-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 9, 328 Seiten

Reihe: Algebra and Applications

The aim of this monograph is to give a self-contained introduction to the modern theory of finite transformation semigroups with a strong emphasis on concrete examples and combinatorial applications. It covers the following topics on the examples of the three classical finite transformation semigroups: transformations and semigroups, ideals and Green's relations, subsemigroups, congruences, endomorphisms, nilpotent subsemigroups, presentations, actions on sets, linear representations, cross-sections and variants. The book contains many exercises and historical comments and is directed first of all to both graduate and postgraduate students looking for an introduction to the theory of transformation semigroups, but also to tutors and researchers.

Autoren/Hrsg.

Ganyushkin, Olexandr

Mazorchuk, Volodymyr

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;Ordinary and Partial Transformations;13
3.1;1.1 Basic Definitions;13
3.2;1.2 Graph of a (Partial) Transformation;15
3.3;1.3 Linear Notation for Partial Transformations;20
3.4;1.4 Addenda and Comments;22
3.5;1.5 Additional Exercises;25
4;The Semigroups Tn, PT n,and ISn;27
4.1;2.1 Composition of Transformations;27
4.2;2.2 Identity Elements;29
4.3;2.3 Zero Elements;31
4.4;2.4 Isomorphism of Semigroups;32
4.5;2.5 The Semigroup;34
4.6;2.6 Regular and Inverse Elements;35
4.7;2.7 Idempotents;38
4.8;2.8 Nilpotent Elements;40
4.9;2.9 Addenda and Comments;43
4.10;2.10 Additional Exercises;47
5;Generating Systems;51
5.1;3.1 Generating Systems in T n, PT n, and ISn;51
5.2;3.2 Addenda and Comments;54
5.3;3.3 Additional Exercises;55
6;Ideals and Green’s Relations;56
6.1;4.1 Ideals of Semigroups;56
6.2;4.2 Principal Ideals in T n, PT n, and ISn;57
6.3;4.3 Arbitrary Ideals in T n, PT n, and ISn;61
6.4;4.4 Green’s Relations;64
6.5;4.5 Green’s Relations on T n, PT n, and ISn;69
6.6;4.6 Combinatorics of Green’s Relationsin the Semigroups T n, PT n, and ISn;71
6.7;4.7 Addenda and Comments;73
6.8;4.8 Additional Exercises;76
7;Subgroups and Subsemigroups;79
7.1;5.1 Subgroups;79
7.2;5.2 Cyclic Subsemigroups;80
7.3;5.3 Isolated and Completely Isolated Subsemigroups;84
7.4;5.4 Addenda and Comments;93
7.5;5.5 Additional Exercises;98
8;Other Relations on Semigroups;100
8.1;6.1 Congruences and Homomorphisms;100
8.2;6.2 Congruences on Groups;103
8.3;6.3 Congruences on T n, PT n, and ISn;105
8.4;6.4 Conjugate Elements;112
8.5;6.5 Addenda and Comments;117
8.6;6.6 Additional Exercises;119
9;Endomorphisms;120
9.1;7.1 Automorphisms of T n, PT n, and ISn;120
9.2;7.2 Endomorphisms of Small Ranks;123
9.3;7.3 Exceptional Endomorphism;124
9.4;7.4 Classification of Endomorphisms;127
9.5;7.5 Combinatorics of Endomorphisms;132
9.6;7.6 Addenda and Comments;136
9.7;7.7 Additional Exercises;137
10;Nilpotent Subsemigroups;139
10.1;8.1 Nilpotent Subsemigroups and Partial Orders;139
10.2;8.2 Classification of Maximal Nilpotent Subsemigroups;142
10.3;8.3 Cardinalities of Maximal Nilpotent, Subsemigroups;146
10.4;8.4 Combinatorics of Nilpotent Elements in ISn;149
10.5;8.5 Addenda and Comments;156
10.6;8.6 Additional Exercises;159
11;Presentation;161
11.1;9.1 Defining Relations;161
11.2;9.2 A presentation for ISn;164
11.3;9.3 A Presentation for T n;169
11.4;9.4 A presentation for PT n;177
11.5;9.5 Addenda and Comments;180
11.6;9.6 Additional Exercises;181
12;Transitive Actions;183
12.1;10.1 Action of a Semigroup on a Set;183
12.2;10.2 Transitive Actions of Groups;185
12.3;10.3 Transitive Actions of T n;187
12.4;10.4 Actions Associated with L-Classes;188
12.5;10.5 Transitive Actions of PT n and ISn;190
12.6;10.6 Addenda and Comments;193
12.7;10.7 Additional Exercises;195
13;Linear Representations;197
13.1;11.1 Representations and Modules;197
13.2;11.2 L-Induced S-Modules;200
13.3;11.3 Simple Modules over ISn and PT n;203
13.4;11.4 Effective Representations;206
13.5;11.5 Arbitrary ISn-Modules;208
13.6;11.6 Addenda and Comments;212
13.7;11.7 Additional Exercises;219
14;Cross-Sections;222
14.1;12.1 Cross-Sections;222
14.2;12.2 Retracts;223
14.3;12.3 H-Cross-Sections in T n, PT n, and ISn;226
14.4;12.4 L-Cross-Sections in T n and PT n;229
14.5;12.5 L-Cross-Sections in ISn;233
14.6;12.6 R-Cross-Sections in ISn;238
14.7;12.7 Addenda and Comments;240
14.8;12.8 Additional Exercises;242
15;Variants;244
15.1;13.1 Variants of Semigroups;244
15.2;13.2 Classification of Variants for ISn, T n,and PT n;247
15.3;13.3 Idempotents and Maximal Subgroups;250
15.4;13.4 Principal Ideals and Green’s Relations;252
15.5;13.5 Addenda and Comments;253
15.6;13.6 Additional Exercises;256
16;Order-Related Subsemigroups;258
16.1;14.1 Subsemigroups, Related to the Natural Order;258
16.2;14.2 Cardinalities;260
16.3;14.3 Idempotents;264
16.4;14.4 Generating Systems;267
16.5;14.5 Addenda and Comments;274
16.6;14.6 Additional Exercises;280
17;Answers and Hints to Exercises;283
18;Bibliography;289
19;List of Notation;302
20;Index;311