On the monoid of partial isometries of a finite star graph

TL;DR

This paper studies the algebraic structure of the monoid of all partial isometries on a finite star graph, focusing on its rank, presentation, Green's relations, and cardinality.

Contribution

It provides the first detailed analysis of the monoid of partial isometries on star graphs, including rank, presentation, and Green's relations.

Findings

Determined the rank of the monoid $DPS_n$.

Provided a presentation for $DPS_n$.

Calculated the cardinality of $DPS_n$.

Abstract

In this paper we consider the monoid $DPS_n$ of all partial isometries of a star graph $S_n$ with $n$ vertices. Our main objectives are to determine the rank and to exhibit a presentation of $DPS_n$. We also describe Green's relations of $DPS_n$ and calculate its cardinal.