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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/
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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:

(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

Information

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

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References

REFERENCES

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