On disjunction of equations in the semigroup language with no constants

TL;DR

This paper proves that in the language of semigroups without constants, no nontrivial semigroup can be an equational domain, meaning finite unions of algebraic sets are not algebraic in such structures.

Contribution

It establishes a fundamental limitation on the algebraic structure of semigroups in the standard language, showing they cannot be equational domains.

Findings

No nontrivial semigroup is an equational domain in the standard semigroup language.

Finite unions of algebraic sets are not algebraic in nontrivial semigroups.

The result clarifies the algebraic properties of semigroups without constants.

Abstract

A semigroup $S$ is an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove that every nontrivial semigroup in the standard language $\{\cdot\}$ is not an equational domain.