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Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 203–237 | Cite as

Interpretable groups in Mann pairs

  • Haydar Göral
Article

Abstract

In this paper, we study an algebraically closed field \(\Omega \) expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \(\Gamma \). This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple \((\Omega , k, \Gamma )\). This enables us to characterize the interpretable groups when \(\Gamma \) is divisible. Every interpretable group H in \((\Omega ,k, \Gamma )\) is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in \(\Gamma \) by an interpretable group N, which is the quotient of an algebraic group by a subgroup \(N_1\), which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in \(\Gamma \).

Keywords

Model theory Stability Mann pairs 

Mathematics Subject Classification

03C45 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsKoç UniversitySarıyerTurkey

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