Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T05:53:52.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable definability and generic relations

Published online by Cambridge University Press:  12 March 2014

Byunghan Kim  and
Rahim Moosa
Byunghan Kim
Affiliation:
Yonsei University, Department of Mathematics, 134 Shinchon-Dong, Seodaemun-Gu Seoul 120-749, Korea. E-mail: bkim@yonsei.ac.kr
Rahim Moosa
Affiliation:
University of Waterloo, Department of Pure Mathematics 200 University Avenue West Waterloo, Ontario N2L 3G1, Canada. E-mail: rmoosa@math.uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

An amalgamation base p in a simple theory is stably definable if its canonical base is interde-finable with the set of canonical parameters for the ϕ-definitions of p as ϕ ranges through all stable formulae. A necessary condition for stably definability is given and used to produce an example of a supersimple theory with stable forking having types that are not stably definable. This answers negatively a question posed in [8]. A criterion for and example of a stably definable amalgamation base whose restriction to the canonical base is not axiomatised by stable formulae are also given. The examples involve generic relations over non CM-trivial stable theories.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1163 - 1176
Copyright
Copyright © Association for Symbolic Logic 2007

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]Baudisch, A. and Pillay, A., A free pseudospace, this Journal, vol. 65 (2000), no. 1, pp. 443–460.Google Scholar
[2]Chatzidakis, Z. and Pillay, A., Generic structures and simple theories, Annals of Pure and Applied Logic, vol. 95 (1998), no. 1-3, pp. 71–92.CrossRefGoogle Scholar
[3]Duret, J., Les corps faiblement algébriquement dos non séparablement dos ont la proprieté d'indépendence, Model theory of algebra and arithmetic (Pacholski, L., Wierzejewski, J., and Wilkie, A.J., editors), Lecture notes in mathematics, vol. 834, Springer-Verlag, 1980, pp. 136–160.CrossRefGoogle Scholar
[4]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), no. 2, pp. 147–166.CrossRefGoogle Scholar
[5]Hrushovski, E., Psuedo-finite fields and related structures, Model theory and applications (Bélair, L., Chatzidakis, Z., D'Aquino, P., Marker, D., Otero, M., Point, F., and Wilkie, A., editors), quaderni di matematica, vol. 11, Seconda Università di Napoli, 2005, pp. 151–212.Google Scholar
[6]Kim, B., Simplicity, and stability in there, this Journal, vol. 66 (2001), no. 2, pp. 822–836.Google Scholar
[7]Kim, B.., Hart, B., and Pillay, A., Coordinatisation and canonical bases in simple theories, this Journal, vol. 65 (2000), no. 1, pp. 293–309.Google Scholar
[8]Kim, B. and Pillay, A., Around stable forking, Fundamenta Mathematicae, vol. 170 (2001), no. 1-2, pp. 107–118.CrossRefGoogle Scholar
[9]Pillay, A., A note on CM-triviality and the geometry of forking, this Journal, vol. 65 (2000), no. 1, pp. 474–480.Google Scholar
[10]Wagner, F., Relational structures and dimensions, Automorphisms of first-order structures, Oxford University Press, 1994, pp. 153–180.CrossRefGoogle Scholar