Natural partial orders on transformation semigroups with fixed sets (Q1751446)

Summary: Let \(X\) be a nonempty set. For a fixed subset \(Y\) of \(X\), let \(\operatorname{Fix} \left(X, Y\right)\) be the set of all self-maps on \(X\) which fix all elements in \(Y\). Then \(\operatorname{Fix} \left(X, Y\right)\) is a regular monoid under the composition of maps. In this paper, we characterize the natural partial order on \(\operatorname{Fix}(X, Y)\) and this result extends the result due to \textit{G. Kowol} and \textit{H. Mitsch} [Monatsh. Math. 102, 115--138 (1986; Zbl 0594.20062)]. Further, we find elements which are compatible and describe minimal and maximal elements.