**Mathematical Surveys and Monographs**

Volume: 46;
1997;
166 pp;
Hardcover

MSC: Primary 20;
Secondary 05

**Print ISBN: 978-0-8218-0627-2
Product Code: SURV/46**

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

**Electronic ISBN: 978-1-4704-1277-7
Product Code: SURV/46.E**

List Price: $63.00

AMS Member Price: $50.40

MAA Member Price: $56.70

# Symmetric Inverse Semigroups

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*Stephen Lipscomb*

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.

#### Readership

Graduate students and research mathematicians working in semigroup theory. Also of interest to computer scientists looking for a guide into areas of original research in semigroups.

#### Reviews & Endorsements

A most welcome addition to the literature of semigroup theory. The existence of a standard reference and a standard notation should ensure that further work on symmetric inverse semigroups will take place in a more organized and coherent way than has hitherto been possible.

-- Bulletin of the London Mathematical Society

An enthusiastic account, full of detail, worked examples, and pictures. Researchers in transformation semigroups will find it an accessible and useful book to dip into for facts and for ideas.

-- Zentralblatt MATH

For most of us who are interested in semigroups, this text will be a really profitable surprise!

-- Monatshefte für Mathematik

The book is notable for a great number of examples, tableaux, diagrams, figures, and other illustrations that help the reader to grasp the subject and give him food for exercises. The historical remarks and comments are very useful.

-- Semigroup Forum

#### Table of Contents

# Table of Contents

## Symmetric Inverse Semigroups

- Contents ix10 free
- Preface xiii14 free
- Introduction xv16 free
- Chapter 1. Decomposing Charts 120 free
- Chapter 2. Basic Observations 928
- Chapter 3. Commuting Charts 1938
- Chapter 4. Centralizers of Permutations 2544
- Chapter 5. Centralizers of Charts 3352
- Chapter 6. Alternating Semigroups 4362
- Chapter 7. S[sub(n)]-normal Semigroups 5372
- Chapter 8. Normal Semigroups and Congruences 6988
- Chapter 9. Presentations of Symmetric Inverse Semigroups 83102
- Chapter 10. Presentations of Alternating Semigroups 97116
- Chapter 11. Decomposing Partial Transformations 107126
- §50 Semigroup Hierarchy 107126
- §51 Path Notation for Partial Transformations 108127
- §52 Cilia and Cells of Partial Transformations 111130
- §53 Idempotents and Nilpotents 112131
- §54 Multiplication of Partial Transformations 112131
- §55 Cyclic Semigroups of Partial Transformations 114133
- §56 Comments 115134

- Chapter 12. Commuting Partial Transformations 117136
- Chapter 13. Centrahzers, Conjugacy, Reconstruction 129148
- §62 Centrahzers 129148
- §63 Conjugacy in C[sub(n)] 130149
- §64 Conjugacy in PT[sub(n)] 133152
- §65 S[sub(n)]-normal Semigroups in PT[sub(n)] 133152
- §66 Graph Reconstruction 133152
- §67 Isomorphic Categories of Graphs and Semigroup Extensions 136155
- §68 Semigroup Reconstruction Conjecture 139158
- §69 Comments 140159

- Appendix 141160
- §70 Objects and Elements 141160
- §71 Groups and Group Morphisms 142161
- §72 Permutation Groups 143162
- §73 Centrahzers of Permutations as Direct Products 146165
- §74 Centrahzers of Regular Permutations as Wreath Products 147166
- §75 Semigroups 148167
- §76 Semigroup Morphisms and Congruences 150169
- §77 Green's Relations 151170
- §78 Free Semigroups, Free Monoids, and Free Groups 152171
- §79 Categories 152171

- Bibliography 155174
- Index 163182