On three submonoids of the dihedral inverse monoid on a finite set

In this paper we consider three submonoids of the dihedral inverse monoid ${\mathcal{D}\mathcal{I}}_n$, namely its 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 all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green’s relations and determine the rank.