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Groupoids, covers, and 3-uniqueness in stable theories

Published online by Cambridge University Press:  12 March 2014

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

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
Copyright
Copyright © Association for Symbolic Logic 2010

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References

[1]Chatzidakis, Zoé and Hrushovski, Ehud, Model theory of difference fields, Transactions of the American Mathematical Society, vol. 351 (1999), pp. 29973071.CrossRefGoogle Scholar
[2]de Piro, Tristram, Kim, Byunghan, and Millar, Jessica, Constructing the type-definable group from the group configuration, Journal of Mathematical Logic, vol. 6 (2006), pp. 121139.CrossRefGoogle Scholar
[3]Evans, David, Finite covers with finite kernels, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 109147.CrossRefGoogle Scholar
[4]Hrushovski, Ehud, Unidimensional theories are superstable, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 117138.CrossRefGoogle Scholar
[5]Hrushovski, Ehud, Groupoids, imaginaries and internal covers, preprint. arXiv: math. LO/0603413.Google Scholar
[6]Pillay, Anand, Geometric stability theory, Oxford University Press, 1996.Google Scholar
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