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On one-based theories

Published online by Cambridge University Press:  12 March 2014

E. Bouscaren  and
E. Hrushovski
E. Bouscaren
Affiliation:
Université Paris7, CNRS URA 753-UFR de Mathématiques, 75251 Paris Cedex 05France
E. Hrushovski
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Department of Mathematics, Hebrew University, Jerusalem 91904, Israel
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Extract

We know from [H1], [H2] that in a stable theory, given a nontrivial locally modular regular type q, one can define a group with generic domination equivalent to q, and that the dependence relation on q can be analyzed in terms of this group. In a stable one-based theory, every regular type is locally modular; hence, this result holds for every nontrivial regular type. We show here that, in fact, in a stable one-based theory, a similar type of construction can be done without the assumption of regularity. More precisely, we show that for any type q, the nontrivial part of q can be analyzed by generics of groups and that any nontrivial relation can be described by affine relations (Theorem A).

This construction is then used to answer a question about homogeneity in pairs of models which is still open in the case of arbitrary stable theories (Theorem C).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 2 , June 1994 , pp. 579 - 595
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[B1] Bouscaren, E., The group configuration—after E. Hrushovski, The model theory of groups (1989) (Nesin, A. and Pillay, A., editors), Notre Dame Mathematical Lectures, vol. 11, University of Notre Dame Press, Notre Dame, Indiana, 1989.Google Scholar
[B2] Bouscaren, E., Dimensional order property and pairs of models, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 205–231.Google Scholar
[Fi] Fisher, E., Abelian structures I, Abelian groups theory, 2nd New Mexico State University Conference 1976, Lecture Notes in Mathematics, vol. 616, Springer-Verlag, Berlin and New York.Google Scholar
[H1] Hrushovski, E., Contributions to model theory, Ph.D. thesis , University of California at Berkeley, Berkeley, California, 1986.Google Scholar
[H2] Hrushovski, E., Locally modular regular types, Classification theory: Chicago 1985, Springer-Verlag, New York, 1987.Google Scholar
[HP] Hrushovski, E. and Pillay, A., Weakly normal groups, Proceedings of Logic Colloquium '85, Paris, North-Holland, Amsterdam, 1986.Google Scholar
[Pi] Pillay, A., Stable theories, pseudoplanes and the number of countable models, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 147–160.Google Scholar
[Pr] Prest, M., Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge and New York, 1988.Google Scholar