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∩, 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, modelcompleteness, 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.