Algebraic geometry over groups III: Elements of model theory

One of the main results of this paper is that elementary theories of coordinate groups Γ(Yi) of irreducible components Yi of an algebraic set Y over a group G are interpretable in the coordinate group Γ(Y) of Y for a wide class of groups G. This implies, in particular, that one can study model theory of Γ(Y) via the irreducible coordinate groups Γ(Yi). This result is based on the technique of orthogonal systems of subdirect products of domains, which we develop here. It has some other interesting applications, for example, if H is a finitely generated group from the quasi-variety generated by a free non-abelian group F, then H is universally equivalent either to a unique direct product Fl of l copies of F or to the group Fl×Z, where Z is an infinite cyclic.