Diameters of endomorphism monoids of chains

TL;DR

This paper calculates the topological invariants called diameters for the endomorphism monoid of chains, revealing their values depend on whether the chain is finite or infinite and its specific structure.

Contribution

It provides explicit computations of the left and right diameters of endomorphism monoids of chains, a novel analysis of these invariants in this context.

Findings

Infinite chains have left diameter 2 and right diameter 2 or 3.

Finite chains have endomorphism monoids with diameters 0 or 1.

Right diameter equals 2 or 3 depending on chain quotient structure.

Abstract

The left and right diameters of a monoid are topological invariants defined in terms of suprema of lengths of derivation sequences with respect to finite generating sets for the universal left or right congruences. We compute these parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically, if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a quotient of $C{\setminus}\{z\}$ for some endpoint $z$. If $C$ is finite then so is $End(C),$ in which case the left and right diameters are 1 (if $C$ is non-trivial) or 0.