A partial order on transformation semigroups with restricted range that preserve double direction equivalence

Let be the full transformation semigroup on a set . For an equivalence on , let For each nonempty subset of , we denote the restriction of to by . Let be the intersection of the semigroup with the semigroup of all transformations with restricted range under the condition that . Equivalently, , where . Then is a subsemigroup of . In this paper, we characterize the natural partial order on . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.