Journal of Symbolic Logic

Definability of groups in ℵ₀-stable metric structures

Itaï Ben Yaacov

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Abstract

We prove that in a continuous ℵ₀-stable theory every type-definable group is definable. The two main ingredients in the proof are:

1. Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from [Ben08], allowing us to prove the theorem in case the metric is invariant under the group action; and

2. Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 3 (2010), 817-840.

Dates
First available in Project Euclid: 9 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1278682202

Digital Object Identifier
doi:10.2178/jsl/1278682202

Mathematical Reviews number (MathSciNet)
MR2723769

Zentralblatt MATH identifier
1205.03047

Subjects
Primary: 03C45: Classification theory, stability and related concepts [See also 03C48] 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)

Keywords
Continuous logic definable set definable group definable metric ℵ₀-stability

Citation

Ben Yaacov, Itaï. Definability of groups in ℵ₀-stable metric structures. J. Symbolic Logic 75 (2010), no. 3, 817--840. doi:10.2178/jsl/1278682202. https://projecteuclid.org/euclid.jsl/1278682202


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