Automorphism groups of centralizers of idempotents

Abstract

For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ ρ, consider the following subsemigroup of the semigroup T( X) of full transformations on X: T(X,ρ,R)= a∈T(X): Ra⊆R and (x,y)∈ρ⇒(xa,ya)∈ρ . The semigroup T( X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and image R. We prove that the automorphisms of T( X, ρ, R) are the inner automorphisms induced by the units of T( X, ρ, R) and that the automorphism group of T( X, ρ, R) is isomorphic to the group of units of T( X, ρ, R).