Mathematical Surveys and Monographs 1997; 166 pp; hardcover Volume: 46 ISBN10: 0821806270 ISBN13: 9780821806272 List Price: US$63 Member Price: US$50.40 Order Code: SURV/46
 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. 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 "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
