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Symmetric Inverse Semigroups
Stephen Lipscomb, Mary Washington College, Fredericksburg, VA

Mathematical Surveys and Monographs
1997; 166 pp; hardcover
Volume: 46
ISBN-10: 0-8218-0627-0
ISBN-13: 978-0-8218-0627-2
List Price: US$63
Member Price: US$50.40
Order Code: SURV/46
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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 (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.


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.


"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

  • Decomposing charts
  • Basic observations
  • Commuting charts
  • Centralizers of permutations
  • Centralizers of charts
  • Alternating semigroups
  • \(S_n\)-normal semigroups
  • Normal semigroups and congruences
  • Presentations of symmetric inverse semigroups
  • Presentations of alternating semigroups
  • Decomposing partial transformations
  • Commuting partial transformations
  • Centralizers, conjugacy, reconstruction
  • Appendix
  • Bibliography
  • Index
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