On semigroups of orientation-preserving partial permutations with restricted range

Let ${\mathrm{\Omega }}_n$ be a finite chain with $n$ elements $(n\in \mathbb{N})$, and let ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{I}}_n$ be the semigroup of all injective orientation-preserving partial transformations of ${\mathrm{\Omega }}_n$. In this paper, for any nonempty subset $Y$ of ${\mathrm{\Omega }}_n$, we consider the subsemigroup ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{I}}_n(Y)$ of ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{I}}_n$ of all transformations with range contained in $Y$. We describe the Green’s relations and study the regularity of ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{I}}_n(Y)$. Moreover, we calculate the rank of ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{I}}_n(Y)$ and determine when two semigroups of this type are isomorphic.