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

  • Montserrat Casals-Ruiz (a1) and Ilya V. Kazachkov
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.

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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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