On disjunctions of equations over semigroups

TL;DR

This paper investigates the conditions under which semigroups with finite ideals are equational domains, meaning finite unions of algebraic sets are algebraic, advancing understanding of algebraic structures over semigroups.

Contribution

It provides necessary and sufficient conditions for semigroups with finite ideals to be equational domains, a novel characterization in algebraic semigroup theory.

Findings

Characterization of equational domains among semigroups with finite ideals

Necessary and sufficient conditions identified

Enhanced understanding of algebraic sets over semigroups

Abstract

A semigroup $S$ is called an equational domain (e.d.) if any finite union of algebraic sets over $S$ is algebraic. For a semigroup $S$ with a finite ideal we find the necessary and sufficient conditions to be an e.d.