Ranks of monoids of endomorphisms of a finite undirected path

TL;DR

This paper investigates the minimal generating sets (ranks) of monoids formed by endomorphisms and weak endomorphisms of finite undirected paths, providing new insights into their algebraic structure.

Contribution

It determines the ranks of monoids of all endomorphisms and weak endomorphisms of finite undirected paths, including strong variants, from a generator perspective.

Findings

Calculated the ranks of monoids of endomorphisms of P_n.

Determined the ranks of monoids of weak endomorphisms of P_n.

Analyzed strong and strong weak endomorphisms of P_n.

Abstract

In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak endomorphisms and all endomorphisms of the undirected path $P_n$ with $n$ vertices. We also consider strong and strong weak endomorphisms of $P_n$.