Interpolative Fusions

We define the interpolative fusion of multiple theories over a common reduct, a notion that aims to provide a general framework to study model-theoretic properties of structures with randomness. In the special case where the theories involved are model complete, their interpolative fusion is precisely the model companion of their union. Several theories of model-theoretic interest are shown to be canonically bi-interpretable with interpolative fusions of simpler theories. We initiate a systematic study of interpolative fusions by also giving general conditions for their existence and relating their properties to those of the individual theories from which they are built.