Generic expansions of countable models

Barbina, Silvia and Zambella, Domenico (2012). Generic expansions of countable models. Notre Dame Journal of Formal Logic, 53(4) pp. 511–523.

DOI: https://doi.org/10.1215/00294527-1722728

URL: http://projecteuclid.org/euclid.ndjfl/1352383229

Abstract

We compare two different notions of generic expansions of countable saturated structures. One kind of genericity is related to existential closure, 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,σ 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. We prove that when all existentially closed models of T₀ have the same existential theory, (N,σ) is Truss generic if and only if (N,σ) is an e-atomic model. When T is ω-categorical and T₀ has a model companion T_mc, the e-atomic models are simply the atomic models of T_mc.

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