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Preprint Number 1375
1375. Elías Baro and Amador Martín-Pizarro Open core and small groups in dense pairs of topological structures E-mail: Submission date: 26 January 2018. Revised 13 November 2019 (with new title and abstract). Abstract: Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate. Mathematics Subject Classification: 03C64, 03C45 Keywords and phrases: |
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