TY - JOUR
T1 - Algebraic geometry over groups III
T2 - Elements of model theory
AU - Kvaschuk, Alexei
AU - Myasnikov, Alexei
AU - Remeslennikov, Vladimir
PY - 2005/6/1
Y1 - 2005/6/1
N2 - 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.
AB - 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.
KW - Algebraic geometry
KW - Definability
KW - Groups
KW - Irreducible components
KW - Model theory
KW - Quasi-varieties
UR - http://www.scopus.com/inward/record.url?scp=17844374631&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=17844374631&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2004.07.038
DO - 10.1016/j.jalgebra.2004.07.038
M3 - Article
AN - SCOPUS:17844374631
SN - 0021-8693
VL - 288
SP - 78
EP - 98
JO - Journal of Algebra
JF - Journal of Algebra
IS - 1
ER -