Groups That Together with Any Transformation Generate Regular Semigroups or Idempotent Generated Semigroups

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈G, a 〉 \G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates ag = g−1ag of a by elements g ∈ G generate a semigroup denoted 〈ag | g ∈ G〉. We classify the finite permutation groups G on a finite set X such that the semigroups 〈G, a〉, 〈G, a〉\G, and 〈ag | g ∈ G〉 are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups 〈G, a 〉 \G and 〈 ag | g ∈ G 〉 are generated by their idempotents for all non-invertible transformations of X. Date: 30 October 2009