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Tame expansions of \({\omega}\)-stable theories and definable groups. (English) Zbl 1529.03199

Summary: We study groups definable in tame expansions of \({\omega}\)-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair \((K,G)\), where \(K\) is an algebraically closed field and \(G\) is a multiplicative subgroup of \(K^{\times}\) with the Mann property, we show that the pair \((K,G)\) satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in \((K,G)\) in terms of algebraic groups in \(K\) and interpretable groups in \(G\). Furthermore, we compute the Morley rank and the U-rank in \((K,G)\) and both ranks agree.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C60 Model-theoretic algebra
20F11 Groups of finite Morley rank

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