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


Published:20100318
Printed: Jun 2010
Montserrat CasalsRuiz,
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
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 quantifierfree formulas over a nonabelian 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 firstorder 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, 03C10, 20F06 show english descriptions
Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] Quantifier elimination, model completeness and related topics Cancellation theory; application of van Kampen diagrams [See also 57M05]
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]
