  Let ${\cal PO}_n$ 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 ${\cal PO}_n$.
Several consequences of these bijections were discussed.
