On the Zariski topology of $\mathrm{\Omega }$-groups

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