On Presentations of Semigroup of Transformations Restricted by an Equivalence

Abstract

For a non-empty set X denote the full transformation semigroup of X by T(X). Let \(\sigma\) be an equivalence relation on X and E(X, \(\sigma\)) denotes the semigroup (under composition) of all \(\alpha\) : X \(\mapsto\) X, such that \(\sigma\) \(\subseteq\) ker(\(\alpha\) ). Semigroup of transformations with restricted equivalence occur when we take all transformations whose kernel is contained in some fixed equivalence, E(X, \(\sigma\)). First, we found that E(X, \(\sigma\)) is a disjoint union copies of two generating sets. Next, we discuss the presentations, acts, subacts, direct products and bilateral semidirect product of the semigroup of transformation with restricted equivalence E(X, \(\sigma\)) and its application.