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 St:=〈G,t〉 can be written as a product eg, where e2=e∈St 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.