Partial Automorphisms and Injective Partial Endomorphisms of a Finite Undirected Path

TL;DR

This paper investigates the structure of partial automorphisms and injective partial endomorphisms of finite undirected paths, providing formulas for their monoid ranks, Green's relations, and cardinalities from a semigroup theory perspective.

Contribution

It offers explicit formulas for the ranks and detailed structural descriptions of the monoids of partial automorphisms and endomorphisms of finite paths.

Findings

Formulas for the ranks of $IEnd(P_n)$ and $PAut(P_n)$

Descriptions of Green's relations for these monoids

Calculations of their cardinalities

Abstract

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $IEnd(P_n)$ and $PAut(P_n)$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $P_n$ with $n$ vertices. We also describe Green's relations of $PAut(P_n)$ and $IEnd(P_n)$ and calculate their cardinals.