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In this paper we consider three submonoids of the dihedral inverse monoid$\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$and $\mathcal{ODI}_n$ of all orientation-preserving, monotone andorder-preserving transformations, respectively. For each of these threemonoids, we compute the cardinal, give descriptions of Green's relations anddetermine the rank.

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