Primitive permutation groups and strongly factorizable transformation semigroups

Abstract

Let Ω be a finite set and T (Ω) be the full transformation monoid on Ω. The rank of a transformation t ∈ T (Ω) is the natural number |Ωt|. Given A ⊆ T (Ω), denote by A the semigroup generated by A. Let k be a fixed natural number such that 2 ≤ k ≤ |Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformations t ∈ T (Ω), every element in S t := G, t can be written as a product eg, where e 2 = e ∈ S t and g ∈ G. In the second part we prove, among other results, that if S ≤ T (Ω) and G is the normalizer of S in the symmetric group on Ω, then the semigroup SG is regular if and only if S is regular. (Recall that a semigroup S is regular if for all s ∈ S there exists s ∈ S such that s = ss s.) The paper ends with a list of problems.

A transversal property for permutation groups motivated by partial transformations

“…Let A ⊆ T (Ω), the full transformation monoid on Ω; classify the permutation groups G ≤ S n such that G, a is regular for all a ∈ A. For many different sets A, this problem has been considered in [2,3,4,8,10,19,20,21,25], among others. The goal of this paper is to consider the similar problem when A is a set of partial transformations with prescribed domain and image sizes (Theorem 1.6).…”

Sync-Maximal Permutation Groups Equal Primitive Permutation Groups

“…See [7] for a good survey and definitions. Furthermore, permutation groups in general have been investigated with respect to the properties of resulting transformation monoid if non-permutations were added [4,5,6,7].…”