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Groupoids, covers, and 3-uniqueness in stable theories

Published online by Cambridge University Press:  12 March 2014

John Goodrick  and
Alexei Kolesnikov
John Goodrick
Affiliation:
Department of Mathematics, Mathematics Building, University of Maryland, College Park, Md 20742, USA. E-mail: goodrick@math.umd.edu
Alexei Kolesnikov
Affiliation:
Department of Mathematics, Towson University, Towson, Md 21252. USA. E-mail: akolesnikov@towson.edu
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Abstract

Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 905 - 929
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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