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Open core and small groups in dense pairs of topological structures

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2022

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Baro E, Martin-Pizarro A. Open core and small groups in dense pairs of topological structures. Annals of Pure and Applied Logic 2021;172:102858. https://doi.org/10.1016/j.apal.2020.102858.

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

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