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

Published online by Cambridge University Press:  20 November 2018

Montserrat Casals-Ruiz  and
Ilya V. Kazachkov
Montserrat Casals-Ruiz
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 2K6, e-mail: casalsruiz@math.mcgill.ca, kazachkov@math.mcgill.ca
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Abstract

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

Keywords

20F1003C1020F06
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 3 , 01 June 2010 , pp. 481 - 519
Copyright
Copyright © Canadian Mathematical Society 2010

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