Natural Partial Orders on Transformation Semigroups with Fixed Sets

Abstract

LetXbe a nonempty set. For a fixed subsetYofX, letFixX,Ybe the set of all self-maps onXwhich fix all elements inY. ThenFixX,Yis a regular monoid under the composition of maps. In this paper, we characterize the natural partial order onFix(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.