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.