On pomonoid of partial transformations of a poset (Q6611511)

For a poset \((X,\leq)\), let \(\mathcal{PT}(X)\) denote the monoid of all partial transformations and \(\mathcal{PI}(X)\) denote the submonoid of \(\mathcal{PT}(X)\) of all injective transformations on \(X\). A transformation \(f\in \mathcal{PT}(X)\) is called \emph{order-preserving} if \(x\leq y\) implies \(xf\leq yf\) for all \(x,y\in \mathrm{dom}(f)\), and \(f\) is called \emph{order-embedding} if \(x\leq y\) if and only if \(xf\leq yf\), for all \(x,y\in \mathrm{dom}(f)\). Moreover, let \(\mathcal{PO}(X)\) (\(\mathcal{POE}(X)\)) denote the submonoid of \(\mathcal{PT}(X)\) of all order-preserving transformations and \(\mathcal{POE}(X)\) denote the submonoid of \(\mathcal{PT}(X)\) of all order-embedding transformations on \(X\). For any \(f\in \mathcal{PT}(X)\), the upper closures of \(\mathrm{dom}(f)\) and \(\mathrm{im}(f)\) are defined by \[\begin{array}{rcl} \mathrm{dom}(f)^{\uparrow}&=&\{ x\in X : \exists y\in \mathrm{dom}(f),\,\, y\leq x\} \mbox{ and}\\\N\mathrm{im}(f)^{\uparrow}&=&\{ x\in X : \exists y\in \mathrm{im}(f),\,\, y\leq x\}, \end{array}\] respectively. Then let \[\begin{array}{rcl} \mathcal{PO}^{\uparrow}(X)&=&\{ f\in \mathcal{PO}(X) : \mathrm{dom}(f)^{\uparrow}=\mathrm{dom}(f)\} \mbox{ and}\\\N\mathcal{POE}^{\uparrow}(X)&=&\{ f\in \mathcal{POE}(X) : \mathrm{im}(f)^{\uparrow}=\mathrm{im}(f)\}. \end{array}\]\N\NNow let \((X,\leq)\) be a poset and equip \(\mathcal{PO}(X)\) with the following point-wise order relation: For all \(f,g\in \mathcal{PO}(X)\), \[f\leq g \quad \Leftrightarrow \quad \mathrm{dom}(f)\subseteq \mathrm{dom}(g) \mbox{ and }\forall x\in \mathrm{dom}(f),\,\, xf\leq xg.\] In the second section, it is shown with an example that \((\mathcal{PO}(X),\leq)\) is not a pomonoid in general. However, it is shown in Theorem 2.4 that \((\mathcal{PO}^{\uparrow}(X),\leq)\) is a pomonoid. In Section 3, the author shows that \(\mathcal{POE}^{\uparrow}(X)\) is also a submonoid of \(\mathcal{POE}(X)\) and that \(\mathcal{IPO}^{\uparrow}(X)=\mathcal{PO}^{\uparrow}(X)\cap \mathcal{POE}^{\uparrow}(X)\) is a subpomonoid of \(\mathcal{PO}^{\uparrow}(X)\). It is also shown that \(\mathcal{IPO}^{\uparrow}(X)\) is an inverse pomonoid. In Section 4, it is shown that if \(X\) is a finite totally ordered set, then \(\mathcal{IPO}^{\uparrow}(X)\) is (weakly) right reversible and, moreover, \(\mathcal{IPO}^{\uparrow}(X)\) a right and left unitary in \(\mathcal{PO}^{\uparrow}(X)\).