Algebraic geometry over groups. III: Elements of model theory. (Q555757)

[For part II cf. \textit{A. Myasnikov} and \textit{V. Remeslennikov}, ibid. 234, No. 1, 225-276 (2000; Zbl 0970.20017).] The authors develop an approach to direct products of domains via orthogonal systems (of idempotents) similar to the classical one in the ring theory. This approach is applied in their proof that the elementary theories of coordinate groups \(\Gamma(Y_i)\) of irreducible components \(Y_i\) of an algebraic set \(Y\) over a group \(G\) are interpretable in the coordinate group \(\Gamma(Y)\) of \(Y\) for a wide class of groups \(G\), containing every equationally Noetherian, non-Abelian CSA-group. There are some other interesting applications of the authors' methods. In particular, every finitely generated group \(H\) from the quasi-variety generated by a free non-Abelian group \(F\) is universally equivalent either to a unique direct product \(F^l\) of \(l\) copies of \(F\) or to the group \(F^l\times\mathbb{Z}\), where \(\mathbb{Z}\) is infinite cyclic.