Lattice Paths and Order-preserving Partial Transformations

Let ${\mathcal{P}\mathcal{O}}_n$ be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that there exist bijections between the set of certain lattice paths in the Cartesian plane that start at $(0,0)$, end at $(n-1,n-1)$, and certain subsemigroups of ${\mathcal{P}\mathcal{O}}_n$. Several consequences of these bijections were discussed.