On the Zariski topology of $\Omega$-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.