On the Zariski topology of $\Omega$-groups

TL;DR

This paper investigates the geometric properties of $ abla$-groups within a specific variety, characterizing when such groups are equational domains based on algebraic variety unions.

Contribution

It provides necessary and sufficient conditions for $ abla$-groups to be equational domains in their variety, linking geometric properties to algebraic conditions.

Findings

Characterization of when an $ abla$-group is an equational domain

Conditions for the union of algebraic varieties to be algebraic

Insights into the geometric structure of $ abla$-groups

Abstract

A number of geometric properties of $\Omega$-groups from a given variety of $\Omega$-groups can be characterized using the notions of domain and equational domain. An $\Omega$-group $H$ of a variety $\Theta$ is an equational domain in $\Theta$ if the union of algebraic varieties over $H$ is an algebraic variety. We give necessary and sufficient conditions for an $\Omega$-group $H$ in $\Theta$ to be an equational domain in this variety.