Open core and small groups in dense pairs of topological structures

TL;DR

This paper investigates the structure of dense pairs of geometric topological fields, showing they have tame open core and that definable functions and groups are closely related to the reduct, with implications for model-theoretic properties.

Contribution

It fixes a gap in van den Dries's work and proves that definable functions and groups in dense pairs are essentially definable in the reduct, extending understanding of their model-theoretic structure.

Findings

Definable open sets in dense pairs are already definable in the reduct.

Unary definable functions agree with reduct functions outside small definable sets.

Groups internal to the predicate have locally definable group laws in the reduct.

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.