Definable and invariant types in enrichments of NIP theories

TL;DR

This paper investigates conditions under which definable and invariant types in enriched NIP theories retain their definability and invariance, extending prior work and applying results to valued differential fields.

Contribution

It provides a sufficient condition for types in enriched NIP theories to be definable or invariant, generalizing previous results and applying to valued differential fields.

Findings

Conditions for definability and invariance in enriched theories

Generalization of density of definable types among non-forking types

Application to valued differential fields

Abstract

Let T be an NIP L-theory and T' be an enrichment. We give a sufficient condition on T' for the underlying L-type of any definable (respectively invariant) type over a model of T' to be definable (respectively invariant) as an L-type. Besides, we generalise work of Simon and Starchenko on the density of definable types among non forking types to this relative setting. These results are then applied to Scanlon's model completion of valued differential fields.