Nullstellensatz for relative existentially closed groups

TL;DR

This paper establishes a Nullstellensatz for $G$-existentially closed elements within varieties of $G$-groups, generalizing previous theorems and revealing structural properties of these elements.

Contribution

It generalizes Theorem G to all varieties of $G$-groups, showing that $G$-existentially closed elements satisfy Nullstellensatz for finite systems and relate to quasi-varieties.

Findings

$G$-existentially closed elements satisfy Nullstellensatz.

Pairs of such elements generate the same quasi-variety.

If both are $q_{ ext{omega}}$-compact, they are geometrically equivalent.

Abstract

We prove that in every variety of $G$-groups, every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize {\bf Theorem G} of \cite{BMR1}. As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{\omega}$-compact, they are geometrically equivalent.