On the Rank of Monoids of Endomorphisms of a Finite Directed Path

TL;DR

This paper investigates the algebraic structure of the monoid of weak endomorphisms of a finite directed path, determining its rank, describing regular elements, and counting idempotents, thus extending understanding of order-preserving transformations.

Contribution

It provides the first determination of the rank of the monoid of weak endomorphisms of a finite directed path and characterizes its regular elements and idempotent count.

Findings

Calculated the rank of the monoid of weak endomorphisms.

Described the regular elements within the monoid.

Counted the number of idempotents in the monoid.

Abstract

In this paper we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid $\wEnd\vec{P}_n$ of all weak endomorphisms of a directed path with $n$ vertices, which is a submonoid of the widely studied monoid $\On$ of all order-preserving transformations of a $n$-chain. Also, we describe the regular elements of $\wEnd\vec{P}_n$ and calculate its size and number of idempotents.