Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T21:27:40.107Z Has data issue: false hasContentIssue false
  • English
  • Français

A rank for the class of elementary submodels of a superstable homogeneous model

Published online by Cambridge University Press:  12 March 2014

Tapani Hyttinen  and
Olivier Lessmann
Tapani Hyttinen
Affiliation:
Department of Mathematics, University of Helsinki, P.O. Box 4, 00014, Finland, E-mail: tapani.hyttinen@helsinki.fi
Olivier Lessmann
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK, E-mail: lessmann@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any dependence relation for superstable homogeneous models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 4 , December 2002 , pp. 1469 - 1482
Copyright
Copyright © Association for Symbolic Logic 2002

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]Baldwin, J., Fundamentals of Stability Theory, Springer-Verlag, Berlin, 1988.CrossRefGoogle Scholar
[2]Berenstein, A., Generic expansions of Hilbert spaces, preprint.Google Scholar
[3]Berenstein, A. and Buechler, S., Homogeneous expansions of Hilbert spaces, preprint.Google Scholar
[4]Buechler, S. and Lessmann, O., Simple homogeneous models, Journal of the American Mathematical Society, to appear.Google Scholar
[5]Cherlin, G., Groups ofsmall Morley rank, Annals of Mathematical Logic, vol. 17 (1979), pp. 1–28.CrossRefGoogle Scholar
[6]Djordjevic, M., Finite variable logic, stability, and finite models, this Journal, vol. 66 (2001), no. 2, pp. 837–858.Google Scholar
[7]Grossberg, R. and Lessmann, O., Shelah's stability and homogeneity spectrum in finite diagrams, Archive for Mathematical Logic, vol. 41 (2002), no. 1, pp. 1–31.CrossRefGoogle Scholar
[8]Hrushovski, E., Simplicity and the Lascar group, preprint.Google Scholar
[9]Hrushovski, E., Unimodular minimal structures, Journal of the London Mathematical Society, vol. 46 (1992), pp. 385–394.CrossRefGoogle Scholar
[10]Hyttinen, T., Canonical finite diagrams and quantifier elimination, preprint.Google Scholar
[11]Hyttinen, T., ω-saturated models which omit countable types, Technical Report 90, Department of Mathematics, University of Helsinki, 1995, preprint, pp. 1–20.Google Scholar
[12]Hyttinen, T., Generalizing Morley's theorem, Mathematical Logic Quarterly, vol. 44 (1998), pp. 176–184.CrossRefGoogle Scholar
[13]Hyttinen, T., On non-structure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematicae, vol. 156 (1998), pp. 167–182.CrossRefGoogle Scholar
[14]Hyttinen, T. and Lessmann, O., Interpreting groups in strongly minimal homogeneous models, preprint.Google Scholar
[15]Hyttinen, T. and Shelah, S., Main gap for locally saturated elementary submodels of a homogeneous structure, this Journal, to appear.Google Scholar
[16]Hyttinen, T., Strong splitting in stable homogeneous models, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 201–228.CrossRefGoogle Scholar
[17]Iovino, J., Stable Banach spaces and Banach space structures I: Fundamentals, Models, Algebra, and Proofs, Lecture Notes in Pure and Applied Mathematics, vol. 203, Dekker, New York, 1999, pp. 77–95.Google Scholar
[18]Keisler, H. J., Model Theory for Infinitary Logic, North-Holland, Amsterdam, 1971.Google Scholar
[19]Lessmann, O., Ranks and pregeometries in finite diagrams, Annals of Pure and Applied Logic, vol. 106 (2000), no. 1, pp. 49–83.CrossRefGoogle Scholar
[20]Pillay, A., Forking in the category of existentially closed models, preprint.Google Scholar
[21]Pillay, A., Geometric Stability Theory, Oxford Logic Guides, vol. 32, Clarendon Press, Oxford, 1996.CrossRefGoogle Scholar
[22]Shelah, S., Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69–118.CrossRefGoogle Scholar
[23]Shelah, S., A combinatorial problem, order for models and infinitary languages, Pacific Journal of Mathematics, vol. 41 (1972), pp. 247–261.CrossRefGoogle Scholar
[24]Shelah, S., The lazy model theorist's guide to stability, Proceedings of the Symposium “Logique et Analyse” (Louvain, March 1975) (Henrand, P., editor), 1975, pp. 241–308.Google Scholar
[25]Zilber, B., Fields with pseudo-exponentiation, preprint.Google Scholar