NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

Let T-X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup S (X, Y) = {f is an element of T-X : f (Y) subset of Y } of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S (X; Y) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S (X; Y) is abundant.