Automorphism groups of centralizers of idempotents

For a set X, an equivalence relation rho on X, and a cross-section R of the partition X/rho, consider the following subsemigroup of the semigroup T(X) of full transformations on X: T(X, rho, R) = {a is an element of T(X): Ra subset of or equal to R and (x, y) is an element of rho double right arrow (xa, ya) is an element of rho}. The semigroup T(X, rho, R) is the centralizer of the idempotent transformation with kernel rho and image R. We prove that the automorphisms of T(X, rho, R) are the inner automorphisms induced by the units of T(X, rho, R) and that the automorphism group of T(X, rho, R) is isomorphic to the group of units of T(X, rho, R). (C) 2003 Elsevier Inc. All rights reserved.