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

Abstract

Abstract Let T ( X ) T\left(X) be the full transformation semigroup on a set X X . For an equivalence E E on X X , let T E ∗ ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇔ ( x α , y α ) ∈ E } . {T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}. For each nonempty subset Y Y of X X , we denote the restriction of E E to Y Y by E Y {E}_{Y} . Let T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) be the intersection of the semigroup T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) with the semigroup of all transformations with restricted range Y Y under the condition that ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Equivalently, T E ∗ ( X , Y ) = { α ∈ T E ∗ ( X ) : X α ⊆ Y } {T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\} , where ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Then T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) is a subsemigroup of T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) . In this paper, we characterize the natural partial order on T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.