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

Abstract

The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets).Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F , where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Limit groups over partially commutative groups and group actions on real cubings

“…As we already mentioned above, partially commutative groups are equationally Noetherian and hence any finitely generated residually G group is a subdirect product of the direct product of finitely many fully residually G groups; see the authors [13] and [3]. We show that such an embedding can be given effectively.…”

“…By [13,Lemma 4.17;14,Corollary 2.11], the group A (an extension of a centraliser of the canonical generator a) is discriminated by G by this family of homomorphisms. Therefore, we conclude that T 0 is discriminated by G by the family of homomorphisms induced by the solutions that factor through the branch.…”

Algebraic geometry over algebraic structures. IV. Equational domains and codomains

“…In [16], it was proved that a finitely generated partially commutative non-Abelian group is a domain iff it is directly indecomposable. For every n 5, G n is directly indecomposable and, consequently, is a domain.…”

Partially commutative metabelian groups: centralizers and elementary equivalence

“…If the vertex x 4 sits on the path l(x 1 , x 2 ) chosen, then applying ϕ 4 : S Γ → S Γ4 to (13) gives us…”

“…In resent years a series of works have appeared dealing with algebraic geometry over partially commutative groups and with elementary theory for such groups. In [13], for instance, it was proved that a positive theory for a partially commutative group is decidable.…”