Regular, unit-regular, and idempotent elements of semigroups of transformations that preserve a partition

Let X be a nonempty set and T-X the full transformation semigroup on X. For a partition P of X, we study semigroups T(X, P) = {f is an element of T-X vertical bar (for all X-i is an element of P)(there exists X-j is an element of P) X-i f subset of X-j}, Sigma(X, P) = {f is an element of T(X, P) vertical bar (for all X-i is an element of P) Xf boolean AND X-i not equal empty set}, and Gamma(X, P) = {f is an element of T-X vertical bar (for all X-i is an element of P)(there exists X-j is an element of P) Xi f = X-j} under composition. We give necessary and sufficient conditions for Gamma(X, P) to be the semigroup of all closed selfmaps on X endowed with the topology having P as a basis. For finite X, we characterize unit-regular elements of T(X, P) and Sigma(X, P). We discuss set inclusions between Gamma(X, P) and certain semigroups of selfmaps that preserve P. We characterize idempotents and regular elements of Gamma(X, P). For finite X, we also prove that all regular elements of Gamma(X, P) are unit-regular. We finally count the number of elements, idempotents, and regular elements of Gamma(X, P) for finite X.