Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-19T21:10:40.329Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

[1]Chatzidakis, Zoé and Hrushovski, Ehud, Model theory of difference fields, Transactions of the American Mathematical Society, vol. 351 (1999), pp. 2997–3071.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. 121–139.CrossRefGoogle Scholar
[3]Evans, David, Finite covers with finite kernels, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 109–147.CrossRefGoogle Scholar
[4]Hrushovski, Ehud, Unidimensional theories are superstable, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 117–138.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.CrossRefGoogle Scholar