Permutation groups and transformation semigroups: results and problems

Abstract

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics.This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S = G, a such as regularity and generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided.These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.

Congruences on direct products of transformation and matrix monoids

“…Corollary 3. 10. Let (f, g), (f , g ) ∈ Q m × Q n , such that g = g , (g, g ) ∈ H and (f, f ) ∈ H. Let j = max{|f |, |f |} and k = |g| = |g |.…”

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

“…These are polynomials of degrees r + 1, r, r + 1 respectively; so 3r + 2 ≥ r 2 − r, giving r ≤ 4. Now PGL(2, 4) ∼ = A 5 falls under case (a); and computation shows that (PGL (2,9), λ) is closed but PGL(2, 16) and PGL(2, 16) : 2 are not. (Both the last two groups have three orbits on 4-sets, but PGL(2, 16) : 4 has only two.…”

“…So r v − r ≤ 3r − 3, whence r = 2. But the computer establishes that not all PGL(2, 9)-orbits are fixed by PΓL (2,9).…”

“…• G = N Sn (G), that is, • G = AGL (1,8), PGL (2,8), PGL (2,9), M 10 , PSL(2, 11), M 22 , PXL (2,25), or PXL (2,49), with k = 3 and λ = (4, 1, . .…”

Idempotency of Left-Sided Identity on Singular Transformation Semigroup

“…In 1968, Tainiter [6] came up with a formula for knowing the number of idempotent in T n , which he denoted by . Araújo and Cameron [1] obtained that a semigroup S generated by the conjugates g -1 ag with aT n \S n and g  S n , is idempotent generated, regular and that S = A partition of a set S is a pairwise disjoint set of non-empty subsets, called "parts" or "blocks" or "cells", whose union is all of S. The semigroup S = T n \S n as studied in this work, is decomposable into non -overlapping cells using the semigroup aS n . Howie [4] used Sing n to denote singular mappings and proved that Sing n , the semigroup of all singular mappings of X = {1,2, .…”