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

We give a presentation for the monoid \(\mathscr{I}\mathscr{O}_n\) $\mathcal{I}{\mathcal{O}}_n$ of all order-preserving transformations of an n-chain whose ranges are intervals. We also consider the submonoid \(\mathscr{I}\mathscr{O}_n^-\) $\mathcal{I}{\mathcal{O}}_n^{-}$ of \(\mathscr{I}\mathscr{O}_n\) $\mathcal{I}{\mathcal{O}}_n$ consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.