On the ranks of certain semigroups of order-preserving partial isometries of a finite chain

Let $X_n={1,2,ldots,n}$ be a finite chain, $mathcal{ODP}_{n}$ be the semigroup of order-preserving partial isometries on $X_n$ and $N$ be the set of all nilpotents in $mathcal{ODP}_{n}$. In this work, we study the nilpotents in $mathcal{ODP}_{n}$ and investigate the ranks of two subsemigroups of $mathcal{ODP}_{n}$; the nilpotent generatedsubsemigroup $langle Nrangle$ and the subsemigroup ~$L(n,r)= { alpha in mathcal{ODP}_{n} : |im~alpha|leq r}$.