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) of T(X, Y) by T E (X , Y) = { α ∈ T (X , Y) : ∀ (x , y) ∈ E , (x α , y α) ∈ E }. We study 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) and then describe the maximal and minimal elements. [ABSTRACT FROM AUTHOR]