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

Abstract

Let T (X) be the full transformation semigroup on a set X. For an equivalence E on X, let TE∗ (X) = {α ϵ T (X): ∀ x, y ϵ X, (x, y) ϵ E ⇔ (x α, y α) ϵ E }. For each nonempty subset Y of X, we denote the restriction of E to Y by EY. Let TE∗ (X, Y) be the intersection of the semigroup TE∗ (X) with the semigroup of all transformations with restricted range Y under the condition that |X/E| = |Y/EY|. Equivalently, TE∗ (X, Y) = { α ϵ TE∗ (X): X α ⊆ Y }, where |X/E| = |Y/EY|. Then TE∗ (X, Y) is a subsemigroup of TE∗(X). In this paper, we characterize the natural partial order on TE∗ (X, Y). Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of TE∗ (X, Y) lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.