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

Published:2010-03-18
Printed: Jun 2010
• Montserrat Casals-Ruiz,
Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 2K6
• Ilya V. Kazachkov,
Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 2K6
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## Abstract

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 $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.
 MSC Classifications: 20F10 - Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 03C10 - Quantifier elimination, model completeness and related topics 20F06 - Cancellation theory; application of van Kampen diagrams [See also 57M05]

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