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Stable definability and generic relations

Published online by Cambridge University Press:  12 March 2014

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
Corresponding

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
Copyright
Copyright © Association for Symbolic Logic 2007

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References

[1]Baudisch, A. and Pillay, A., A free pseudospace, this Journal, vol. 65 (2000), no. 1, pp. 443460.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. 7192.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. 136160.CrossRefGoogle Scholar
[4]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), no. 2, pp. 147166.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. 151212.Google Scholar
[6]Kim, B., Simplicity, and stability in there, this Journal, vol. 66 (2001), no. 2, pp. 822836.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. 293309.Google Scholar
[8]Kim, B. and Pillay, A., Around stable forking, Fundamenta Mathematicae, vol. 170 (2001), no. 1-2, pp. 107118.CrossRefGoogle Scholar
[9]Pillay, A., A note on CM-triviality and the geometry of forking, this Journal, vol. 65 (2000), no. 1, pp. 474480.Google Scholar
[10]Wagner, F., Relational structures and dimensions, Automorphisms of first-order structures, Oxford University Press, 1994, pp. 153180.Google Scholar

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