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

Document Type : Research Paper

Authors

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

Abstract

Let

X_{n} = {1, 2, \dots, n}

be a finite chain,

{ODP}_{n}

be the semigroup of order-preserving partial isometries on

X_{n}

and

N

be the set of all nilpotents in

{ODP}_{n}

. In this work, we study the nilpotents in

{ODP}_{n}

and investigate the ranks of two subsemigroups of

{ODP}_{n}

; the nilpotent generated
subsemigroup

⟨ N ⟩

and the subsemigroup ~

L (n, r) = {α \in {ODP}_{n} : | i m α | \leq r}

Keywords