Regular centralizers of idempotent transformations

Denote by T (X) the semigroup of full transformations on a set X. For ε ∈ T (X), the centralizer of ε is a subsemigroup of T (X) defined by C(ε) = {α ∈ T (X) : αε = εα}. It is well known that C(idX) = T (X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(idX) contains all non-invertible transformations in C(idX). This paper generalizes this result to C(ε), an arbitrary regular centralizer of an idempotent transformation ε ∈ T (X), by describing the subsemigroup generated by the idempotents of C(ε). As a corollary we obtain that the subsemigroup generated by the idempotents of a regular C(ε) contains all non-invertible transformations in C(ε) if and only if ε is the identity or a constant transformation.