Generic Expansions of Countable Models

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 $T_{\mathrm {mc}}$.