Natural Partial Orders on Transformation Semigroups with Fixed Sets

摘要

Let X be a nonempty set. For a fixed subset Y of X, let FixX,Y be the set of all self-maps on X which fix all elements in Y. Then FixX,Y is a regular monoid under the composition of maps. In this paper, we characterize the natural partial order on Fix(X,Y) and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements.