On the monoid of partial isometries of a wheel graph

TL;DR

This paper investigates the algebraic structure of the monoid of all partial isometries of a wheel graph, focusing on computing its rank, subsemigroup ranks, and describing Green's relations.

Contribution

It determines the rank of the monoid of partial isometries of a wheel graph and analyzes its subsemigroups and Green's relations, providing new algebraic insights.

Findings

Calculated the rank of the monoid $DPW_n$

Determined the ranks of three key subsemigroups

Described Green's relations for $DPW_n$ and subsemigroups

Abstract

In this paper, we consider the monoid $DPW_n$ of all partial isometries of a wheel graph $W_n$ with $n+1$ vertices. Our main objective is to determine the rank of $DPW_n$. In the process, we also compute the ranks of three notable subsemigroups of $DPW_n$. We also describe Green's relations of $DPW_n$ and of its three considered subsemigroups.