Endomorphisms of semigroups of monotone transformations (Q6637175)

Let \(\mathcal{PT}_{n}\), \(\mathcal{T}_{n}\) and \(\mathcal{I}_{n}\) denote the partial transformation monoid, the (full) transformation monoid and the partial injective transformation monoid on the finite chain \[\Omega_{n}=\{1< \ldots <n\}\] respectively. An element \(\alpha \in \mathcal{PT}_{n}\) is called \textit{order-preserving} (\textit{order-reversing}) if \(x\leq y\) implies \(x\alpha\leq y\alpha\) (\(x\alpha\geq y\alpha\)) for all \(x, y\in \mathrm{dom}(\alpha)\), and \(\alpha \in \mathcal{PT}_{n}\) is called \textit{monotone} if it is either order-preserving or order-reversing. \N\NDenote the submonoid of \(\mathcal{PT}_{n}\) consisting of all monotone partial transformations by \(\mathcal{POD}_{n}\), the submonoid of \(\mathcal{T}_{n}\) consisting of all monotone transformations by \(\mathcal{OD}_{n}\), and the submonoid of \(\mathcal{I}_{n}\) consisting of all monotone partial injective transformations by \(\mathcal{PODI}_{n}\).\N\NFor \(n\geq 2\), let \(M\in \{\mathcal{OD}_{n}, \mathcal{POD}_{n}, \mathcal{PODI}_{n}\}\). In Theorem 2.4, necessary and sufficient conditions for any mapping \(\phi : M\rightarrow M\) to be an endomorphism are given. Furthermore, for each \(M\), the number of endomorphisms of \(M\) are given in Corollary 2.6.