Groups that are the union of two semigroups have left-orderable quotients

TL;DR

This paper characterizes groups that are unions of two proper subsemigroups by linking them to the existence of nontrivial left-orderable quotients, revealing a structural property of such groups.

Contribution

It establishes a precise criterion connecting unions of two proper subsemigroups with the existence of a minimal normal subgroup yielding a left-orderable quotient.

Findings

A group is the union of two proper subsemigroups iff it has a nontrivial left-orderable quotient.

Existence of a minimal normal subgroup with a left-orderable quotient when a group is a union of two semigroups.

Provides a structural characterization of groups based on their subsemigroup unions and quotient properties.

Abstract

In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal subgroup $N\unlhd G$ for which $G/N$ is left-orderable and nontrivial.