Generic Expansions of Countable Models

Notre Dame Journal of Formal Logic 53 (4):511-523 (2012)
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Abstract

We compare two different notions of generic expansions of countable saturated structures. One kind of genericity is related to existential closure, and another is 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 $(N,\sigma)$ denote an expansion of $N$ to the signature $L_{0}$ which is a model of some universal theory $T_{0}$ . We prove that when all existentially closed models of $T_{0}$ have the same existential theory, $(N,\sigma)$ is Truss generic if and only if $(N,\sigma)$ is an e-atomic model. When $T$ is $\omega$ -categorical and $T_{0}$ has a model companion $T_{\mathrm {mc}}$ , the e-atomic models are simply the atomic models of $T_{\mathrm {mc}}$

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References found in this work

Generic structures and simple theories.Z. Chatzidakis & A. Pillay - 1998 - Annals of Pure and Applied Logic 95 (1-3):71-92.
Laforte, G., see Downey, R.T. Arai, Z. Chatzidakis & A. Pillay - 1998 - Annals of Pure and Applied Logic 95 (1-3):287.
Model companions of theories with an automorphism.Hirotaka Kikyo - 2000 - Journal of Symbolic Logic 65 (3):1215-1222.
The strict order property and generic automorphisms.Hirotaka Kikyo & Saharon Shelah - 2002 - Journal of Symbolic Logic 67 (1):214-216.
Les beaux automorphismes.Daniel Lascar - 1991 - Archive for Mathematical Logic 31 (1):55-68.

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