The natural partial order on semigroups of transformations with restricted range that preserve an equivalence

Let Y be a nonempty subset of X and T(X, Y) the set of all functions from X into Y. Then T(X, Y) with composition is a subsemigroup of the full transformation semigroup T(X). Let E be a nontrivial equivalence on X. Define a subsemigroup \(T_E(X,Y)\) $T_E(X,Y)$ of T(X, Y) by \(\begin{aligned} T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}. \end{aligned}\) $$\begin{array}{c}\hfill T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E,(x\alpha ,y\alpha )\in E\}.\end{array}$$ We study \(T_E(X,Y)\) $T_E(X,Y)$ with the natural partial order and determine when two elements are related under this order. We also give a characterization of compatibility on \(T_E(X,Y)\) $T_E(X,Y)$ and then describe the maximal and minimal elements.