Unit Regular Elements in Transformation Semigroups and Continuous Transformation Spaces

In the current literature, most works on transformation semigroups have shifted their attention to characterizing regular or unit regular elements in specialized sub-semigroups, such as those preserving partitions, invariant subspaces, or subspace structures. In this paper, the unit regular elements t of the semigroup T(X) of all transformations defined on an arbitrary set X are characterized. An important property of a unit regular element which states that an element t of T(X) is unit regular if and only if there exists a cross section X0 such that X − X0 and X − R(t) are of the same cardinality has been proved. This result reveals that any transformation t on X having finite range is always unit regular and shows that a transformation t in T(X) which is injective (but not surjective) or surjective (but not injective) can never be a unit regular element of T(X). As a consequence, an alternative proof of the fact that T(X) is a unit regular semigroup if and only if X is finite is given. Analogous characterization of unit regular elements of the semigroup space CT(X) of continuous transformations on a Hausdorff topological space X is also analyzed. The result shows that an element t in CT(X) is unit regular if and only if the range R(t) is a closed subset of X, t is a quotient map onto R(t) with kernel of t possessing continuous cross section X0 such that the closures of X − X0 and X − R(t) are homeomorphic which coincides with t on the boundary X0.