Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-22T21:03:06.474Z Has data issue: false hasContentIssue false
  • English
  • Français

Definability of groups in ℵ0-stable metric structures

Published online by Cambridge University Press:  12 March 2014

Itaï Ben Yaacov
Itaï Ben Yaacov*
Affiliation:
Université Claude Bernard – Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 Boulevard Du 11 Novembre 1918, 69622 Villeurbanne Cedex. France URL: http://math.univ-lyonl.fr/~begnac/
Get access Rights & Permissions [Opens in a new window]

Abstract

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

(i) 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

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

Keywords

Continuous logicdefinable setdefinable groupdefinable metricℵ0-stability
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 817 - 840
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ben03]Yaacov, Itaï Ben, Thickness, and a categoric view of type-space functors, Fundamenta Mathematicæ, vol. 179 (2003), pp. 199–224.CrossRefGoogle Scholar
[Ben05]Yaacov, Itaï Ben, Uncountable dense categoricity in cats, this Journal, vol. 70 (2005), no. 3, pp. 829–860.Google Scholar
[Ben08]Yaacov, Itaï Ben, Topometric spaces and perturbations of metric structures, Logic and Analysis, vol. 1 (2008), no. 3–4, pp. 235–272, arXiv:0802.4458.CrossRefGoogle Scholar
[BenlO]Yaacov, Itaï Ben, Stability and stable groups in continuous logic, this Journal, vol. 75 (2010), no. 3, pp. 1111–1136.Google Scholar
[BBHU08]Yaacov, Itaï Ben, Berenstein, Alexander. Henson, C. Ward, and Usvyatsov, Alexander, Model theory for metric structures, Model theory with applications to algebra and analysis, volume 2 (Chatzidakis, Zoé, Macpherson, Dugald, Pillay, Anand, and Wilkie, Alex, editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, 2008, pp. 315–427.CrossRefGoogle Scholar
[BU]Yaacov, Itaï Ben and Usvyatsov, Alexander, Continuous first order logic and local stability, Transactions of the American Mathematical Society, to appear, arXiv:0801.4303.Google Scholar
[Ber06]Berenstein, Alexander, Definable subgroups of measure algebras, Mathematical Logic Quarterly, vol. 52 (2006), no. 4, pp. 367–374.CrossRefGoogle Scholar