Interpolative fusions II: Preservation results

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 $T_{\cup }^{\ast }$. When each $T_i$ is model-complete, $T_{\cup }^{\ast }$ 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 $T_{\cup }^{\ast }$. 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.

Some parts of this paper originally appeared in the first version of the first interpolative fusion paper. arXiv admin note: text overlap with arXiv:1811.06108