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Preprint Number 1238

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1238. Alexander Berenstein and Evgueni Vassilev
Fields with a dense-codense linearly independent multiplicative subgroup

Submission date: 13 June 2017


We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \mathbb{S}(R), satisfying a density/codensity condition (in the sense of geometric theories). Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the ACF and RCF cases, the resulting theory is near model complete and the expansion preserves many nice model theoretic conditions related to the complexity of definable sets such as stability and NIP. We also analyze the groups definable in the expansion.

Mathematics Subject Classification: 03C45, 03C64

Keywords and phrases: stable theories, NIP theories, unary predicate expansions, definable groups, Mann property

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