On the monoid of partial order-preserving transformations of a finite chain whose domains and ranges are intervals

In this paper, we consider the monoid ${\mathcal{P}\mathcal{I}\mathcal{O}}_n$, of all partial order-preserving transformations on a chain with $n$ elements whose domains and ranges are intervals, along with its submonoid ${\mathcal{P}\mathcal{I}\mathcal{O}}_n^{-}$ of order-decreasing transformations. Our main aim is to give presentations for ${\mathcal{P}\mathcal{I}\mathcal{O}}_n^{-}$ and ${\mathcal{P}\mathcal{I}\mathcal{O}}_n$. Moreover, for both monoids, we describe regular elements and determine their ranks, cardinalities and the numbers of idempotents and nilpotents.