On the existence property over a predicate

TL;DR

This paper proves that in a countable, fully stable theory over a predicate, any complete set can be extended to a model without altering the predicate, demonstrating the Gaifman property and generalizing previous results.

Contribution

It establishes the existence property for countable fully stable theories over a predicate, extending known results to a broader class of theories.

Findings

Complete sets can be extended to models without changing the predicate.

The theory exhibits the Gaifman property, allowing any model of the predicate to be embedded.

Generalizes previous results on stable theories, abelian groups, and difference fields.

Abstract

We prove that in a countable theory T fully stable over a predicate P, any complete set A has the existence property. This means that A can be extended to a model of T without changing the P-part. In particular, T has the Gaifman property: any model of P occurs as the P-part of some model of T. This generalizes results of Lachlan (on stable theories), Hodges (on relatively categorical abelian groups), and Afshordel (on difference fields of characteristic 0).