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 σ is an equivalence relation on X. For every f ∈ T (X) , the kernel of f is defined to be ker f = { (x , y) ∈ X × X ∣ f (x) = f (y) } . Evidently, E (X , σ) = { f ∈ T (X) ∣ σ ⊆ ker f } is a subsemigroup of T(X). Also, the subset R E (X , σ) of E (X , σ) 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 ∼ -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 , σ) . This investigation proves that the semigroup E (X , σ) is always a right Ehresmann semigroup. Finally, we prove that R E (X , σ) is an orthodox semigroup if and only if the set X consists of at most two σ -classes. [ABSTRACT FROM AUTHOR]