On disjunction of equations in inverse semigroups

TL;DR

This paper proves that inverse semigroups that are equational domains in an extended language are necessarily groups, highlighting a fundamental algebraic property.

Contribution

It establishes a new characterization of inverse semigroups as groups when they satisfy the equational domain property in an extended language.

Findings

Inverse semigroups that are equational domains are groups.

The extended language includes multiplication and inverse operations.

The result links algebraic properties to structural classification.

Abstract

A semigroup $S$ is an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove that if an inverse semigroup $S$ is an equational domain in the extended language $\{\cdot,{}^{-1}\}\cup\{s|s\in S\}$ then $S$ is a group.