Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T03:00:11.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Properties and consequences of Thorn-independence

Published online by Cambridge University Press:  12 March 2014

Alf Onshuus
Alf Onshuus*
Affiliation:
Universidad de Los Andes, Departamento de Matemáticas, Cra. Ie No. 18A-10 Edificio h, Bogotá, Colombia. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure.

We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 1 , March 2006 , pp. 1 - 21
Copyright
Copyright © Association for Symbolic Logic 2006

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]Buechler, Steven, Pillay, Anand, and Wagner, Frank, Supersimple theories, Journal of the American Mathematical Society, vol. 14 (2001), no. 1, pp. 109–124.CrossRefGoogle Scholar
[2]Casanovas, Enrique and Wagner, Frank, The free roots of the complete graph, Proceedings of the American Mathematical Society, vol. 132 (2003), no. 5, pp. 1543–1548.CrossRefGoogle Scholar
[3]Hart, Bradd, Kim, Byunghan, and Pillay, Anand, Coordinatisation and canonical bases in simple theories, this Journal, vol. 65 (2000), no. 1, pp. 293–309.Google Scholar
[4]Hodges, Wilfrid, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[5]Kim, Byunghan, Simple first order theories, Ph.D. thesis, University of Notre Dame, 1996.Google Scholar
[6]Kim, Byunghan and Pillay, A., Around stable forking, Fundamenta Mathematicae, vol. 170 (2001), no. 1–2, pp. 107–118, dedicated to the memory of Jerzy Łoś.CrossRefGoogle Scholar
[7]Kim, Byunghan and Pillay, Anand, Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 149–164, joint AILA-KGS Model Theory Meeting (Florence, 1995).CrossRefGoogle Scholar
[8]Pillay, Anand, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996.CrossRefGoogle Scholar
[9]Shelah, S., Classification theory and the number of nonisomorphic models, second ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
[10]van den Dries, Lou, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
[11]Wagner, Frank O., Simple theories, Mathematics and its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar