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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.

Let Y be a fixed subset of a nonempty set X and let F ix(X, Y ) be the set of all self maps on X which fix all elements in Y . Then under the composition of maps, F ix(X, Y ) is a regular monoid. In this paper, we prove that there are only three types of maximal subsemigroups of F ix (X, Y ) and these maximal subsemigroups coincide with the maximal regular subsemigroups when X \ Y is a finite set with |X \ Y | ≥ 2 . We also give necessary and sufficient conditions for F ix(X, Y ) to be factorizable, unit-regular, and directly finite.

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