Interpolative fusions II: Preservation results

TL;DR

This paper investigates conditions under which model-theoretic properties like stability, simplicity, and NIP are preserved when combining theories via interpolative fusion, extending previous results and providing new preservation theorems.

Contribution

It establishes new preservation results for properties such as stability, NIP, and simplicity in the context of interpolative fusion of theories, generalizing prior work and offering sharp examples.

Findings

Quantifier elimination and model-completeness are preserved under certain conditions.

Stability and NSOP1 are preserved when the common reduct is stable.

Simplicity is preserved under stronger algebraic closure hypotheses.

Abstract

We study interpolative fusion, a method of combining theories $T_1$ and $T_2$ in distinct languages in a "generic" way over a common reduct $T_\cap$, to obtain a theory $T_\cup^*$. When each $T_i$ is model-complete, $T_\cup^*$ is the model companion of the union $T_1\cup T_2$. Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of $T_1$ and $T_2$ are inherited by $T_\cup^*$. We first prove preservation results for quantifier elimination, model-completeness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of $T_\cap$, the property $\mathrm{NSOP}_1$ is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in $T_1$ and $T_2$. This generalizes many previous results; for example, simplicity of $\mathrm{ACFA}$ and the random $n$-hypergraph are both non-obvious corollaries. We also address preservation of stability, $\mathrm{NIP}$, and $\aleph_0$-categoricity, and we describe examples which witness that these results are sharp.