Products of idempotents in transformations semigroups

For an idempotent e in the full transformations semigroup on a finite set X, denoted by T(X), the centralizer C(e) is the subsemigroup of T(X) consisting of all elements that commute with e. We have proved that the idempotents of a regular C(e) generate id(X) and all singular transformations in C(e) if and only if C(e)= T(X) or C(e) is isomorphic to P(X') where X' is X with one element removed. It turns out that this result generalizes an important theorem proved by Howie [1966] that says that the idempotents in T(X) generate the identity id(X) and all singular (non invertible) transformations in T(X). A similar result is obtained for P(X), the semigroup of partial transformations on X.