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Hardcover ISBN:  9780821806272 
Product Code:  SURV/46 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470412777 
Product Code:  SURV/46.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821806272 
eBook ISBN:  9781470412777 
Product Code:  SURV/46.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 46; 1997; 166 ppMSC: Primary 20; Secondary 05
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 (oneone 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 4group, 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 Eilenbergstyle 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' idealextension 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.
ReadershipGraduate 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.

Table of Contents

Chapters

1. Decomposing charts

2. Basic observations

3. Commuting charts

4. Centralizers of permutations

5. Centralizers of charts

6. Alternating semigroups

7. $S_n$normal semigroups

8. Normal semigroups and congruences

9. Presentations of symmetric inverse semigroups

10. Presentations of alternating semigroups

11. Decomposing partial transformations

12. Commuting partial transformations

13. Centrahzers, Conjugacy, Reconstruction


Reviews

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


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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 (oneone 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 4group, 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 Eilenbergstyle 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' idealextension 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.
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.

Chapters

1. Decomposing charts

2. Basic observations

3. Commuting charts

4. Centralizers of permutations

5. Centralizers of charts

6. Alternating semigroups

7. $S_n$normal semigroups

8. Normal semigroups and congruences

9. Presentations of symmetric inverse semigroups

10. Presentations of alternating semigroups

11. Decomposing partial transformations

12. Commuting partial transformations

13. Centrahzers, Conjugacy, Reconstruction

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