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

Abstract

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