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Search: MSC category 20F10 ( Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] )

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1. CJM 2010 (vol 62 pp. 481)

Casals-Ruiz, Montserrat; Kazachkov, Ilya V.
 Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups 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. Categories:20F10, 03C10, 20F06

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