Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties are preserved.& COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper provides a complete characterization of the expansion of a complete o-minimal theory by a unary predicate that selects a divisible, dense, and codense group and determines whether it has a model companion. The motivation for this result comes from recent works on the existence of model companions and preservation results for neostability properties. The paper includes examples in which the predicate is an additive subgroup of a real ordered vector space and a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a discussion on the lack of preservation for certain properties when passing to the model companion.