Natural partial order in semigroups of transformations with invariant set

摘要

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 ∈ T_X: f(Y) ? 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.