Interpolative Fusions I

Abstract

We define the interpolative fusion $T_{\cup }^{\ast }$ of a family $(T_i{)}_{i\in I}$ of first-order theories over a common reduct $T_{\cap }$, a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each $T_i$ is model-complete, $T_{\cup }^{\ast }$ coincides with the model companion of $T_{\cup }={\bigcup }_{i\in I}{T}_i$. By obtaining sufficient conditions for the existence of $T_{\cup }^{\ast }$, we develop new tools to show that theories of interest have model companions.

Final version, as published in J. Mathematical Logic. This is about half of the first version of this paper, the other half is now in the second paper