On the monoid of partial isometries of a cycle graph

TL;DR

This paper studies the algebraic structure of the monoid of all partial isometries on a cycle graph, providing a presentation, analyzing Green's relations, and computing its size and rank.

Contribution

It introduces a presentation for the monoid of partial isometries of a cycle graph and characterizes its algebraic properties and structure.

Findings

Presented a presentation of the monoid PC_n.

Described Green's relations for PC_n.

Calculated the cardinality and rank of PC_n.

Abstract

In this paper we consider the monoid $\DPC_n$ of all partial isometries of a $n$-cycle graph $C_n$. We show that $\DPC_n$ is the submonoid of the monoid of all oriented partial permutations on a $n$-chain whose elements are precisely all restrictions of the dihedral group of order $2n$. Our main aim is to exhibit a presentation of $\DPC_n$. We also describe Green's relations of $\DPC_n$ and calculate its cardinal and rank.