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INTERPRETABLE GROUPS, STABLY EMBEDDED SETS, AND VAUGHTIAN PAIRS

Published online by Cambridge University Press:  08 August 2003

BERNHARD HERWIG ,
EHUD HRUSHOVSKI  and
DUGALD MACPHERSON
BERNHARD HERWIG
Affiliation:
Am Gonsenheimer Spiess 18, 55122 MainzGermany e-mail: herwig@math.uni-freiburg.de
EHUD HRUSHOVSKI
Affiliation:
Department of Mathematics, Hebrew University, Jerusalem, Israel e-mail: ehud@math.huji.ac.il
DUGALD MACPHERSON
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT e-mail: h.d.macpherson@leeds.ac.uk
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Abstract

The paper concerns sufficiently saturated structures M over a countable language with a unary predicate P. It is shown that if $P(M)$ is stably embedded and there are no Vaughtian pairs with respect to P, then an infinite group is interpretable over M (in an infinitary sense of ‘interpretable’). Also, it is shown that if M is $\omega$-categorical, $f\,{:}\,D\,{\longrightarrow}\,P$ is a 0-definable map with finite fibres, and $P(M)$ is stably embedded but D is not, then some infinite group is first-order interpretable over M.

Keywords

03C52
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 68 , Issue 1 , August 2003 , pp. 1 - 11
Copyright
The London Mathematical Society 2003

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Footnotes

The research of the first and third authors was partially supported by EPSRC grant GR/K60503. This research was conducted by Ehud Hrushovski as a CMI Prize Fellow, for the Clay Mathematics Institute.