Generic Expansions of Countable Models

We compare two different notions of generic expansions of countable saturated structures. On one hand there is a kind of genericity related to modelcompanions and to amalgamation constructions à la Hrushovski-Fräıssé; on the other, there is a notion of generic expansions defined via topological properties and Baire category theory. The second type of genericity was first formulated by Truss for automorphisms. We work with a later generalization, due to Ivanov, to finite tuples of predicates and functions. Let N be a countable saturated model of some complete theory T , and let (N, σ) denote an expansion of N to the signature L0 which is a model of some universal theory T0. Let Trich be the theory of some suitably defined generic (or rich, in our terminology) model, obtained from the amalgamation class naturally associated to T0. We prove that (N, σ) is Truss-generic if and only if (N, σ) is an e-atomic model of Trich. When T is ω–categorical and Trich is model-complete, the e-atomic models are simply the atomic models of Trich. Mathematics Subject Classification 03C10 (primary), 20B27, 03C50 (secondary) ∗The first author gratefully acknowledges support by the Commission of the European Union under contract MEIF-CT-2005-023302 ‘Reconstruction and generic automorphisms’.