On disjunction of equations in completely regular semigroups

TL;DR

This paper investigates the conditions under which a completely regular semigroup is an equational domain, concluding that such semigroups must be completely simple.

Contribution

It establishes that completely regular semigroups are equational domains only if they are completely simple, providing a new characterization in semigroup theory.

Findings

Completely regular semigroups are equational domains only if they are completely simple.

The paper characterizes the structure of semigroups related to algebraic sets and unions.

Provides a necessary condition for a semigroup to be an equational domain.

Abstract

A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove if a completely regular semigroup $S$ is an equational domain then $S$ is completely simple.