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

Document Type : Research Paper


1 Department of Mathematics, Nigeria Defence Academy, Kano, Nigeria

2 Department of Mathematics, Bayero University Kano, Kano, Nigeria

3 Department of Mathematics, Faculty of Physical Sciences, Bayero University, Kano, Nigeria


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 generated
subsemigroup $\langle N\rangle$ and the subsemigroup ~$L(n,r)= \{ \alpha \in \mathcal{ODP}_{n} : |im~\alpha|\leq r\}$.