Green’s Relations and Regularity for Semigroups of Transformations with Restricted Range that Preserve Double Direction Equivalence Relations: Utsithon Chaichompoo, Kritsada Sangkhanan - Team, Support (Thai Journal of Mathematics, 2019)
Let T(X) be the full transformation semigroup on a set X. For an equivalence E on X and a nonempty subset Y of X, let T_E^*(X,Y)={ αT(X):XαY and x,yX,(x,y)E(xα,yα)E}. In this article, we give a necessary and sufficient condition for T_E^*(X,Y) to be a subsemigroup of T(X) under the composition of functions and study the regularity of T_E^*(X,Y). Finally, we characterize Green's relations on this semigroup.