On semigroups of endomorphisms of a chain with restricted range

TL;DR

This paper characterizes the largest regular subsemigroup and Green's relations of endomorphisms of chains with restricted range, providing structural insights and isomorphism criteria for finite cases.

Contribution

It extends the understanding of endomorphism semigroups by describing the largest regular subsemigroup and Green's relations for chains with restricted ranges, including isomorphism and rank results.

Findings

Largest regular subsemigroup characterized

Green's relations on endomorphism semigroups described

Isomorphism conditions and rank calculations for finite chains

Abstract

Let $X$ be a finite or infinite chain and let $O(X)$ be the monoid of all endomorphisms of $X$. In this paper, we describe the largest regular subsemigroup of $O(X)$ and Green's relations on $O(X)$. In fact, more generally, if $Y$ is a nonempty subset of $X$ and $O(X,Y)$ the subsemigroup of $O(X)$ of all elements with range contained in $Y$, we characterize the largest regular subsemigroup of $O(X,Y)$ and Green's relations on $O(X,Y)$. Moreover, for finite chains, we determine when two semigroups of the type $O(X,Y)$ are isomorphic and calculate their ranks.