## Symmetric Inverse SemigroupsWith 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. |

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### 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 |

163 | |

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### Common terms and phrases

1-paths 5n-normal semigroups alternating semigroup Brandt semigroup cell Chapter characterization charts of rank cilia cilium circuits commuting charts congruence congruence on Cn conjugacy classes conjugate contains Corollary cycle notation cyclic semigroup define denote diagram digraph domain egg-box elements exists extends Figure finite follows free inverse monoid G Cn G PTn G Sn graph group theory H-class ideal idempotents idempotents of rank implies induction initial segments isomorphism join representation kernel normal Klein 4-group Lemma Let length maximal proper path morphism multiplication Mx/p nilpotent normal semigroup normal subgroup normal subsemigroup one-one partial transformations path decomposition path notation path structure pfpr Proposition Let quotient Reconstruction Conjecture regular permutation regular semigroup relations Semigroup Forum semilattice subcell subsemigroup subsemigroups of Cn subset suppose symmetric group symmetric inverse semigroup terminal segment transpositions trivial group wreath product