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Fields with a dense-codense linearly independent multiplicative subgroup

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Abstract

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 algebraically closed field and real closed field 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.

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Authors and Affiliations

  1. Universidad de los Andes, Cra 1 No 18A-10, Bogotá, Colombia

    Alexander Berenstein

  2. Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL, A2H 6P9, Canada

    Evgueni Vassiliev

Authors
  1. Alexander Berenstein
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  2. Evgueni Vassiliev
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Correspondence to Alexander Berenstein.

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The authors would like to thank Pantelis Eleftheriou, Haydar Göral, Ayhan Günaydin and Alf Onshuus for several helpful coversations. The first author was partially supported by Colciencias’ project Teoría de modelos y dinámicas topológicas number 120471250707. The second author was partially supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant AP05134992, Conservative extensions, countable ordered models, and closure operators). The authors thank the referee for helpful comments and valuable suggestions.

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Berenstein, A., Vassiliev, E. Fields with a dense-codense linearly independent multiplicative subgroup. Arch. Math. Logic 59, 197–228 (2020). https://doi.org/10.1007/s00153-019-00683-w

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  • DOI: https://doi.org/10.1007/s00153-019-00683-w

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