Nullstellensatz for relative existentially closed groups

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  Theorem G of [J‎. ‎Algebra,  219 (1999) ‎16--79]‎. ‎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.