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 | Ω t | . 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 = s s ′ s .) The paper ends with a list of problems.