Taking model-complete cores

TL;DR

This paper investigates the properties and existence of model-complete core theories, showing that many model-theoretic properties are preserved under core companion formation, with specific classes of theories not closed under this operation.

Contribution

It proves the existence of core companions for -categorical theories and demonstrates the preservation of key properties like stability and NIP, while identifying classes not closed under core companions.

Findings

Core companions always exist for -categorical theories.

Stability, NIP, simplicity, NSOP are preserved in core companions.

Classes of theories interpretable over () and () are not closed under core companions.

Abstract

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; the core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, they always exist for $\omega$-categorical theories. We show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved by moving to the core companion of a theory. On the other hand, we show that the classes of theories of structures interpretable over $({\mathbb N};=)$ and over $({\mathbb Q};<)$ are both not closed under taking core companions. The first class is contained in the class of theories of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal.