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

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