For an arbitrary permutation σ in the semigroup Tn of full transformations on a set with n elements, the regular elements of the centralizer C(σ) of σ in Tn are characterized and criteria are given for C(σ) to be a regular semigroup, an inverse semigroup, and a completely regular semigroup.
Second centralizers of partial transformations on a finite set are determined. In particular, it is shown that the second centralizer of any partial transformation $\alpha $ consists of partial transformations that are locally powers of $\alpha $.