Symmetric Inverse Semigroups

Front Cover
American Mathematical Soc., Jan 1, 1996 - Mathematics - 166 pages
0 Reviews
With over 60 figures, tables, and diagrams, this text is both an intuitive introduction to and a rigorous study of finite symmetric inverse semigroups. The model, denoted $C_n$, consists of all charts (one-one partial transformations) of the set ${1,\dots,n}$ under the usual composition of mappings. It has the symmetric groups $S_n$ as a subgroup, and many classical features of $S_n$ are extended to $C_n$. It turns out that these semigroups enjoy many of the classical features of finite symmetric groups. For example, cycle notation, conjugacy, commutativity, parity of permutations, alternating subgroups, Klein 4-group, Ruffini's result on cyclic groups, Moore's presentations of the symmetric and alternating groups, and the centralizer theory of symmetric groups are extended to more general counterparts in $C_n$. Lipscomb classifies normal subsemigroups and also illustrates and applies an Eilenberg-style wreath product. The basic $C_n$ theory is further extended to partial transformation semigroups, and the Reconstruction Conjecture of graph theory is recast as a Rees' ideal-extension conjecture. This books presents much of the material on the theory of finite symmetric inverse semigroups, unifying the classical finite symmetric group theory with its semigroup analogue. A comment section at the end of each chapter provides historical perspective. New proofs, new theorems and the use of multiple figures, tables, and diagrams to present complex ideas make this book current and highly readable.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Decomposing Charts
1
Basic Observations
9
Commuting Charts
19
Centralizers of Permutations
25
Centralizers of Charts
33
Alternating Semigroups
43
25 Members of Acn
44
26 Generators of Alternating Semigroups
47
Decomposing Partial Transformations
107
51 Path Notation for Partial Transformations
108
52 Cilia and Cells of Partial Transformations
111
53 Idempotents and Nilpotents
112
55 Cyclic Semigroups of Partial Transformations
114
56 Comments
115
Commuting Partial Transformations
117
58 Mapping Initial Segments and Cells
118

27 Comments
50
5nnormal Semigroups
53
Semigroups
56
30 Basic Observations
57
31 Semilattices and Ideals
58
32 Nonidempotents in 5nnormal Semigroups
59
33 Classification for n 5
65
35 Comments
66
Normal Semigroups and Congruences
69
37 Congruences on Finite Semilattices
71
38 Kernels of Congruences
75
39 Kernels and Normal Semigroups
77
40 Counting Congruences
80
41 Comments
81
Presentations of Symmetric Inverse Semigroups
83
43 Technical Lemma
90
44 Inductive Lemma
94
45 Comments
96
Presentations of Alternating Semigroups
97
47 Acn Technical Lemma
98
48 Acn Inductive Lemma
103
49 Comments
105
59 Orders of Centralizers of Idempotents
123
60 Howies Theorem
124
61 Comments
126
Centralizers Conjugacy Reconstruction
129
63 Conjugacy in Cn
130
64 Conjugacy in PTn
133
67 Isomorphic Categories of Graphs and Semigroup Extensions
136
68 Semigroup Reconstruction Conjecture
139
69 Comments
140
Appendix
141
71 Groups and Group Morphisms
142
72 Permutation Groups
143
73 Centralizers of Permutations as Direct Products
146
74 Centralizers of Regular Permutations as Wreath Products
147
75 Semigroups
148
76 Semigroup Morphisms and Congruences
150
77 Greens Relations
151
78 Free Semigroups Free Monoids and Free Groups
152
Bibliography
155
Index
163
Copyright

Common terms and phrases

References to this book

Bibliographic information