Algebraic geometry over groups III: Elements of model theory

Abstract

One of the main results of this paper is that elementary theories of coordinate groups Γ (Y i ) of irreducible components Y i 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 (Y i ). 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 F l of l copies of F or to the group F l × Z, where Z is an infinite cyclic.  2004 Elsevier Inc. All rights reserved. (A. Kvaschuk), alexeim@att.net (A. Myasnikov), remesl@iitam.omsk.net.ru (V. Remeslennikov). URL: http://home.att.net/~alexeim/index.htm (A. Myasnikov).

Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups

“…Proof. Since the direct product of two non-abelian groups is never a domain, see [1,18], the result follows immediately from Corollary 4.15.…”

“…Corollary A (cf. [18]). Let G be a partially commutative group and let G = G 1 × · · · × G n × Z r , where G i is a non-abelian directly indecomposable partially commutative group, i = 1, .…”

Limit groups over coherent right-angled Artin groups

“…All torsion-free abelian groups are BP, as is the direct product of a BP group with a torsion-free abelian group. From [26], groups G and H are BP if and only if the free product G * H is BP. However, if G and H are non-abelian groups then the direct product G × H is not a BP group.…”

Изотипные Алгебры

“…Настоящая статья принадлежит к целому ряду статей об универсальной алгебраической геометрии (см. [4], [6]- [9], [15], [25] и др.). Как мы упомянули, в построениях рассматриваемой теории возникают различные вопросы, близкие к алгебре и теории моделей.…”