Algebraic geometry over groups III: Elements of model theory

Alexei Kvaschuk, Alexei Myasnikov, Vladimir Remeslennikov

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)78-98
Number of pages21
JournalJournal of Algebra
Volume288
Issue number1
DOIs
StatePublished - 1 Jun 2005

Keywords

  • Algebraic geometry
  • Definability
  • Groups
  • Irreducible components
  • Model theory
  • Quasi-varieties

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