On expansions of non-abelian free groups by cosets of a finite index subgroup

zbMATH Open1440.03004arXiv1707.03066MaRDI QIDQ4563187

Javier de la Nuez González

Publication date: 6 June 2018

Abstract: Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $qinQ$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarski's problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov's theorem on $forallexists$-sentences and show that the positive theory depends only on $Q$ and neither on the rank of $F$ nor the particular quotient map.

Full work available at URL: https://arxiv.org/abs/1707.03066

Mathematics Subject Classification ID

Model-theoretic algebra (03C60) Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations (03-02) Free nonabelian groups (20E05)

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