On the existence property over a predicate
Alexander Usvyatsov
TL;DR
This work develops classification theory over a distinguished predicate $P$ and proves that for a countable theory $T$ fully stable over $P$, every complete set $A$ has the existence property: there exists $M hd A$ with $P^M=P^A$. Consequently, $T$ has the Gaifman property, and any model of $T^P$ occurs as the $P$-part of some $T$-model. The approach combines a rank-based stability analysis over $P$, completeness criteria, and constructive model-building (local constructions and λ-constructibility) to extend complete sets without altering their $P$-part, yielding full existence. The paper applies these results to ACFA$_0$ (showing full stability over $P$) and discusses nulldimensionality and absolute categoricity over $P$, linking model-theoretic structure to over-$P$ categoricity phenomena. Overall, it unifies and extends Lachlan, Hodges, and Afshordel-type results within a broad, fully stable over $P$ framework, with potential for further applications to fields like exponential-closed fields 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).