Asian Research Journal of Mathematics

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.