On Presentations of Semigroup of Transformations Restricted by an Equivalence

For a non-empty setXdenote the full transformation semigroup ofXbyT(X). Letσbe an equivalencerelation onXandE(X,σ) denotes the semigroup (under composition) of allα:X7−→X, such thatσ⊆ker(α). Semigroup of transformations with restricted equivalence occur when we take all transformationswhose kernel is contained in some fixed equivalence,E(X,σ). First, we found thatE(X,σ) is a disjoint unioncopies of two generating sets. Next, we discuss the presentations, acts, subacts, direct products and bilateralsemidirect product of the semigroup of transformation with restricted equivalenceE(X,σ) and its application.