Abstract
It is shown that the existential theory of \({\mathbb G}\) with rational constraints, over an HNN-extension \({\mathbb G}=\langle {\mathbb H},t; t^{-1}at=\varphi(a) (a \in A) \rangle\) is decidable, provided that the same problem is decidable in the base group \({\mathbb H}\) and that A is a finite group. The positive theory of \({\mathbb G}\) is decidable, provided that the existential positive theory of \({\mathbb G}\) is decidable and that A and ϕ(A) are proper subgroups of the base group \({\mathbb H}\) with A ∩ϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtually-free group is decidable.
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Lohrey, M., Sénizergues, G. (2006). Theories of HNN-Extensions and Amalgamated Products. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4052. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787006_43
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DOI: https://doi.org/10.1007/11787006_43
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