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

Let ${\mathcal{C}\mathcal{P}}_n$ be the semigroup of partial contraction mappings on $[n]=\{1,2,\dots ,n\}$ and let ${\mathcal{O}\mathcal{C}\mathcal{P}}_n$ and ${\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{P}}_n$ 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, ${\mathcal{O}\mathcal{C}\mathcal{P}}_n$ and ${\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{P}}_n$, respectively.