Green's relations on a semigroup of transformations with restricted range that preserves an equivalence relation and a cross-section

Abstract

© 2018 by the authors. 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 a preserves both r (if pa, bq ∈ ρ, then (aα, bα) ∈ ρ) and R (if r P R, then rα ∈ 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 TpX, ρ, Rq when taking ρ to be the identity relation and Y = X, respectively.