Some Results on Semigroups of Transformations Restricted by an Equivalence

For a non-empty set X denote the full transformation semigroup on X by T(X) and suppose that $$\sigma $$ is an equivalence relation on X. For every $$f\in T(X)$$ , the kernel of f is defined to be $$\ker f =\{(x, y)\in X\times X\mid f(x) = f(y)\}$$ . Evidently, $$E(X, \sigma )=\{f\in T(X) \mid \sigma \subseteq \ker f\}$$ is a subsemigroup of T(X). Also, the subset $$RE(X, \sigma )$$ of $$E(X, \sigma )$$ consisting of regular elements is a subsemigroup. Partition of a semigroup by Green’s $$*$$ -relations was first introduced by Fountain in 1979 and the Green’s $$\sim $$ -relations (with respect to a non-empty subset U of the set of idempotents) as a new method of partition were introduced by Lawson (J Algebra 141(2):422–462, 1991). In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup $$E(X, \sigma )$$ . This investigation proves that the semigroup $$E(X, \sigma )$$ is always a right Ehresmann semigroup. Finally, we prove that $$RE(X, \sigma )$$ is an orthodox semigroup if and only if the set X consists of at most two $$\sigma $$ -classes.