Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

Let T ( X , Y ) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X . For an equivalence relation ρ on X , let ρ ^ be the restriction of ρ on Y , R a cross-section of Y / ρ ^ and define T ( X , Y , ρ , R ) to be the set of all total transformations α from X into Y such that α preserves both ρ (if ( a , b ) ∈ ρ , then ( a α , b α ) ∈ ρ ) and R (if r ∈ R , then r α ∈ R ). T ( X , Y , ρ , R ) is then a subsemigroup of T ( X , Y ) . In this paper, we give descriptions of Green’s relations on T ( X , Y , ρ , R ) , and these results extend the results on T ( X , Y ) and T ( X , ρ , R ) when taking ρ to be the identity relation and Y = X , respectively.