The binary quasiorder on semigroups

TL;DR

This paper investigates the properties of the binary quasiorder on semigroups, a relation defined via homomorphisms to the two-element set, revealing its connection to the least semilattice congruence.

Contribution

It introduces new properties of the binary quasiorder and explores its relationship with semilattice congruences in semigroups.

Findings

The binary quasiorder coincides with the least semilattice congruence.

New properties of the binary quasiorder are established.

The paper discusses known and novel aspects of this quasiorder.

Abstract

Given two elements $x,y$ of a semigroup $X$ we write $x\lesssim y$ if for every homomorphism $\chi:X\to\{0,1\}$ we have $\chi(x)\le\chi(y)$. The quasiorder $\lesssim$ is called the $binary$ $quasiorder$ on $X$. It induces the equivalence relation $\Updownarrow$ that coincides with the least semilattice congruence on $X$. In the paper we discuss some known and new properties of the binary quasiorder on semigroups.