Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set

In this paper we consider the submonoids ${\mathcal{O}\mathcal{P}\mathcal{D}\mathcal{I}}_n$, ${\mathcal{M}\mathcal{D}\mathcal{I}}_n$ and ${\mathcal{O}\mathcal{D}\mathcal{I}}_n$ of the dihedral inverse monoid ${\mathcal{D}\mathcal{I}}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids.