Presentations for monoids of finite partial isometries

TL;DR

This paper provides algebraic presentations for the monoid of all partial isometries and its order-preserving submonoid on a finite set, advancing the understanding of their algebraic structure.

Contribution

It introduces explicit presentations for the monoid of finite partial isometries and its order-preserving submonoid, which were not previously characterized.

Findings

Presentations for the monoid of all partial isometries on a finite set.

Presentations for the submonoid of order-preserving partial isometries.

Enhanced understanding of the algebraic structure of these monoids.

Abstract

In this paper we give presentations for the monoid $\mathcal{DP}_n$ of all partial isometries on $\{1,\ldots,n\}$ and for its submonoid $\mathcal{ODP}_n$ of all order-preserving partial isometries.