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The Group Configuration in Simple Theories and its Applications

Published online by Cambridge University Press:  15 January 2014

Itay Ben-Yaacov ,
Ivan Tomašić  and
Frank O. Wagner
Itay Ben-Yaacov
Affiliation:
Équipe de Logique Mathematique, UFR de Mathématiques (Case 7012), Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, FranceE-mail: [email protected]
Ivan Tomašić
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, United KingdomE-mail: [email protected]
Frank O. Wagner
Affiliation:
Institut Girard Desargues, Université Claude Bernard(Lyon 1), Bât. Jean Braconnier, 21 Av. Claude Bernard, 69622 Villeurbanne Cedex FranceE-mail: [email protected]
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Abstract

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity.

The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and the study of polygroups.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 8 , Issue 2 , June 2002 , pp. 283 - 298
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1] Ben-Yaacov, Itay, Group configurations and germs in simple theories, Preprint, 2000.Google Scholar
[2] Ben-Yaacov, Itay, On the fine structure of the polygroup blow-up, Preprint, 2001.Google Scholar
[3] Ben-Yaacov, Itay, Tomašić, Ivan, and Wagner, Frank O., Constructing an almost hyperdefinable group, Preprint, 2001.Google Scholar
[4] Ben-Yaacov, Itay and Wagner, Frank O., Almost orthogonality in simple theories, Preprint, 2001.Google Scholar
[5] Corsini, Piergiulio, Prolegomena of hypergroup theory, Supplement to Rivista di Matematica Pura ed Applicata. Aviani Editore, 1993.Google Scholar
[6] dePiro, Tristam and Kim, Byunghan, The geometry of one-based minimal types, Preprint, 2001.Google Scholar
[7] Evans, David and Wagner, Frank O., Supersimple ω-categorical groups and theories, The Journal of Symbolic Logic, vol. 65(2) (2000), pp. 767–776.Google Scholar
[8] Hrushovski, Ehud, Contributions to stable model theory, Ph.D. thesis , University of California at Berkeley, 1986.Google Scholar
[9] Hrushovski, Ehud, Locally modular regular types, Classification theory: Proceedings of the US-Israel binational workshop on model theory in mathematical logic, Chicago '85 (Baldwin, J., editor), Lecture Notes in Mathematics 1292. Springer Verlag, 1987.Google Scholar
[10] Macpherson, H. D., Absolutely ubiquitous structures and ℵ0-categorical groups, The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 39 (1988), pp. 483–500.CrossRefGoogle Scholar
[11] Pillay, Anand, Geometric stability theory, Clarendon Press, 1996.CrossRefGoogle Scholar
[12] Pillay, Anand, Definability and definable groups in simple theories, The Journal of Symbolic Logic, vol. 63 (1998), pp. 788–796.CrossRefGoogle Scholar
[13] Poizat, Bruno, Groupes stables, Nur al-Mantiq wal-Ma'rifah, 1987.Google Scholar
[14] Shami, Ziv and Wagner, Frank O., On the binding group in simple theories, The Journal of Symbolic Logic, to appear.Google Scholar
[15] Tomašić, Ivan, Geometric simplicity theory, Ph.D. thesis , University of Edinburgh, 2001.Google Scholar
[16] Tomašić, Ivan and Wagner, Frank O., Applications of the group configuration theorem in simple theories, Preprint, 2001.Google Scholar
[17] Vassiliev, Evgueni V., Expansions of rank 1 structures, Ph.D. thesis , University of Notre Dame, 2001.Google Scholar
[18] von Neumann, John, Continuous geometry, Princeton University Press, 1960.Google Scholar
[19] Wagner, Frank O., Stable groups, London Mathematical Society Lecture Notes 240, Cambridge University Press, 1997.Google Scholar
[20] Wagner, Frank O., Simple theories, Kluwer Academic Publishers, 2000.Google Scholar
[21] Wagner, Frank O., Hyperdefinable groups in simple theories, Journal of Mathematical Logic, vol. 1 (2001), pp. 152–172.Google Scholar
[22] Zilber, Boris, Totally categorical theories: Structural properties and the non-finite axiomatizability, Model theory of algebra and arithmetic (Pacholski, L., Wierzejewski, J., and Wilkie, A., editors), Lecture Notes in Mathematics 834. Springer Verlag, 1980, pp. 381–410.Google Scholar
[23] Zilber, Boris, Strongly minimal countably categorical theories III (in Russian), Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 25 (1984), pp. 63–77.Google Scholar