Notre Dame Journal of Formal Logic

Generic Expansions of Countable Models

Silvia Barbina and Domenico Zambella

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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 $T_{\mathrm {mc}}$.

Article information

Source
Notre Dame J. Formal Logic Volume 53, Number 4 (2012), 511-523.

Dates
First available: 8 November 2012

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1352383229

Digital Object Identifier
doi:10.1215/00294527-1722728

Zentralblatt MATH identifier
06111348

Mathematical Reviews number (MathSciNet)
MR2995417

Subjects
Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 20B27: Infinite automorphism groups [See also 12F10] 03C50: Models with special properties (saturated, rigid, etc.)

Keywords
generic automorphism existentially closed structure comeager conjugacy class

Citation

Barbina, Silvia; Zambella, Domenico. Generic Expansions of Countable Models. Notre Dame Journal of Formal Logic 53 (2012), no. 4, 511--523. doi:10.1215/00294527-1722728. http://projecteuclid.org/euclid.ndjfl/1352383229.


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