Stephen Leon Lipscomb

Symmetric inverse semigroups

Decomposing charts Basic observations Commuting charts Centralizers of permutations Centralizers of charts Alternating semigroups

The structure of the centralizer of a permutation

S_n

The alternating semigroups: Generators and congruences

-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.

Presentations of alternating semigroups

Relative to the symmetric groups Sn the structure of centralizers of permutations are known as direct products of certain general wreath products. A recent generalization of the cycle notation for partial one-one transformations (charts) is applied to show that relative to the symmetric inverse semigroups Cn the structure of centralizers of permutations are also direct products of certain subsemigroups of a wreath product, and this latter wreath product includes the former as a subgroup. A necessary and sufficient condition is given for two charts to commute and the approach for the Cn-case parallels and generalizes the one for the Sn-case. As a result, the Cn-case yields the standard known characterizations of commuting permutations, as well as formulas for the orders of centralizers as corollaries. It is an open problem to extend these results to the centralizers of arbitrary charts in Cn.

Classification ofSn-normal semigroups

The idea of an even permutation has recently been generalized, via path notation, to one-one partial transformations (charts) in the symmetric inverse semigroupsCn. The even charts form the alternating semigroupAnc⊂Cn. Generators ofAnc are identified: It then follows that forn≥5,Anc is the collection of restrictions of the even permutations (of rankn). Like theCn case, the congruences ofAnc form a chain.

Centralizers in symmetric inverse semigroups: Structure and order

One presentation of the alternating groupAn hasn−2 generatorss1,…,sn−2 and relationss13=si2=(s1−1si)3=(sjsk)2=1, wherei>1 and |j−k|>1. Against this backdrop, a presentation of the alternating semigroupAnc )An is introduced: It hasn−1 generatorss1,…,Sn−2,e, theAn-relations (above), and relationse2=e, (es1)4, (esj)2=(esj)4,esi=sis1-1es1, wherej>1 andi≥1.

Four-web Graph Paper

IfSn andCn denote, respectively, the symmetric group and inverse semigroup onn symbols, thenSn⊂Cn and a semigroupT⊂Cn isSn-normal ifα−1Tα ⊂Tfor every α∈Sn. TheSn-normal semigroups are classified.

Appendix 3: Inside S 3 and Questions

The representation [5] of the centralizerC(x) of a permutationx in (a symmetric inverse semigroup)Cn involves direct products of wreath products. Indeed, this semigroup case extends its group theory counterpart. Here, the last case (forx nilpotent) is addressed: A quotient of a wreath product is introduced and used to obtain a representation of the correspondingC(x). It follows that, for anyx∈Cn,C(x) can be imbedded in a direct product of wreath products with a quotient of a wreath product. A formula for calculating the order ofC(x) is given. The independent parameters in the formula are precisely those that define the path structure ofx∈Cn.

Appendix 1: Supplement for Chapters 1 and 2

We see what we see because of where we were when. As children our vision sends pictures to our brain, and our brain processes the information so that pictures make sense. We learn to put a square peg in a square hole, and then we move on.

Appendix 2: Supplement for Chapters 3 and 4

This appendix provides an inside view of the 3-sphere and a few thoughts about the limits of physics.