Submission date: 10 January 2022

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_∩, to obtain a theory T_∪^*. When each T_i is model-complete, T_∪^* is the model companion of the union T_1 ∪ 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_∪^*.
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_∩, the property 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 ACFA and the random n-hypergraph are both non-obvious corollaries. We also address preservation of stability, NIP, and ℵ_0-categoricity, and we describe examples which witness that these results are sharp.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2201.03534: pdf, ps.