On the Ranks of Semigroup of Order-preserving or Order-reversing Partial Contraction Mappings on a Finite Chain

Let $\mathcal{CP}_n$ be the semigroup of partial contraction mappings on $[n]=\{1,2,\ldots,n\}$ and let $\mathcal{OCP}_n$ and $\mathcal{ORCP}_n$ be its subsemigroups consisting of all order-preserving and of all order-preserving or order-reversing, partial contraction mappings, respectively. In this paper we obtain the rank of the two semigroups, $\mathcal{OCP}_n$ and $\mathcal{ORCP}_n$, respectively.