Permutable transformation semigroups

then they are simply transitive groups of permutations and centralizers of each other. The problem under consideration has its origin in the theory of networks, [2], where permutability of two transformation semigroups (with respect to a network) refers to an independence relation and hence to concurrency of events in the network. For the results and definitions needed here for the permutation groups we refer to [3]. We shall consider transformation semigroups on a finite set ∆. A subsemigroup S of T∆ is said to be transitive if for all a, b ∈ ∆ there exists an α ∈ S such that α(a) = b . A subsemigroup S of T∆ is a permutation group, if S is a subgroup of the symmetric group Sym(∆). Let C(S) = {β ∈ T∆ | αβ = βα for all α ∈ S}