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Main gap for locally saturated elementary submodels of a homogeneous structure

Published online by Cambridge University Press:  12 March 2014

Tapani Hyttinen  and
Saharon Shelah
Tapani Hyttinen
Affiliation:
Department of Mathematics, P.O. Box 4, 00014, University of Helsinki, Finland, E-mail: [email protected]
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University Jerusalem, Israel Rutgers University, Hill CTR-Bush, New Brunswick, NJ, U.S.A., E-mail: [email protected]
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Abstract

We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1286 - 1302
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Baldwin, J. T., Fundamentals of stability theory, Springer Verlag, 1988.CrossRefGoogle Scholar
[2]Baldwin, J. T. and Shelah, S., The primal framework. I, Annals of Pure and Applied Logic, vol. 46 (1990), pp. 235–264.CrossRefGoogle Scholar
[3]Baldwin, J. T. and Shelah, S., The primal framework. II. smoothness, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 1–34.CrossRefGoogle Scholar
[4]Baldwin, J. T. and Shelah, S., Abstract classes with few models have ‘homogeneous-universal’ models, this Journal, vol. 60 (1995), pp. 246–265.Google Scholar
[5]Grossberg, R., On chains of relatively saturated submodels of a model without the order property, this Journal, vol. 56 (1991), pp. 123–128.Google Scholar
[6]Grossberg, R. and Hart, B., The classification theory of excellent classes, this Journal, vol. 54 (1989), pp. 1359–1381.Google Scholar
[7]Grossberg, R., Iovino, J., and Lessman, O., A primer of simple theories, preprint.Google Scholar
[8]Grossberg, R. and Lessmann, O., Forking in pregeometries, preprint.Google Scholar
[9]Grossberg, R. and Lessmann, O., The main gap for totally transcendental diagrams and abstract decomposition theorems, preprint.Google Scholar
[10]Grossberg, R. and Lessmann, O., Shelah's stability and homogeneity spectrum in finite diagrams, unpublished.Google Scholar
[11]Grossberg, R. and Shelah, S., On the number of nonisomorphic models of an infinitary theory which has the infinitary order property, i, this Journal, vol. 51 (1986), pp. 302–322.Google Scholar
[12]Grossberg, Rami and Shelah, Saharon, On hanf numbers of the infinitary order property, submitted.Google Scholar
[13]Grossberg, Rami and Shelah, Saharon, A nonstructure theorem for an infinitary theory which has the unsuperstability property, Illinois Journal of Mathematics, vol. 30 (1986), pp. 364–390.CrossRefGoogle Scholar
[14]Hart, B. and Shelah, S., Categoricity over P for first order T or categoricity for ϕ ϵ Lω1ω can stop at ℵk while holding for ℵ0, …, ℵk − 1, Israel Journal of Mathematics, vol. 70 (1990), pp. 219–235.Google Scholar
[15]Hrushovski, E., Simplicity and the lascar group, preprint.Google Scholar
[16]Hyttinen, T., Generalizing morley's theorem, Mathematical Logic Quarterly, vol. 44 (1998), pp. 176–184.CrossRefGoogle Scholar
[17]Hyttinen, T., On nonstructure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematicae, vol. 156 (1998), pp. 167–182.CrossRefGoogle Scholar
[18]Hyttinen, T. and Shelah, S., On the number of elementary submodels of an unsuperstable homogeneous structure, Mathematical Logic Quarterly, vol. 44 (1998), pp. 354–358.CrossRefGoogle Scholar
[19]Hyttinen, T., Strong splitting in stable homogeneous models, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 201–228.CrossRefGoogle Scholar
[20]Kim, B. and Pillay, A., From stability to simplicity, The Bulletin of Symbolic Logic, vol. 4 (1998), pp. 17–36.CrossRefGoogle Scholar
[21]Kolman, O. and Shelah, S., Categoricity of theories in Lκω, when n is a measurable cardinal, part I, Fundamenta Mathematicae, vol. 151 (1996), pp. 209–240.Google Scholar
[22]Lascar, D., Stability in model theory, Longman, 1987.Google Scholar
[23]Lessmann, O., Abstract group configuration, preprint.Google Scholar
[24]Lessmann, O., Ranks and pregeometries infinite diagrams, Annals of Pure and Applied Logic, vol. 106 (2000), pp. 49–83.CrossRefGoogle Scholar
[25]Makkai, M. and Shelah, S., Categoricity of theories in Lκω with κ a compact cardinal, Annuls of Pure and Applied Logic, vol. 47 (1990), pp. 41–97.Google Scholar
[26]Pillay, A., Forking in the category of existentially closed structures, unpublished.Google Scholar
[27]Shelah, S., Categoricity in abstract elementary classes: going up inductive steps, to appear.Google Scholar
[28]Shelah, S., Categoricity of an abstract elementary class in two successive cardinals, to appear.Google Scholar
[29]Shelah, S., Categoricity of theories in Lκω, when κ is a measurable cardinal. Part II, to appear.Google Scholar
[30]Shelah, S., Non-structure theory, to appear.Google Scholar
[31]Shelah, S., Universal classes, to appear.Google Scholar
[32]Shelah, S., Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69–118.CrossRefGoogle Scholar
[33]Shelah, S., The lazy model theorist's guide to stability, Logique et Analyze, vol. 71-72 (1975), pp. 241–308.Google Scholar
[34]Shelah, S., Better quasi-orders for uncountable cardinals, Israel Journal of Mathematics, vol. 42 (1982), pp. 177–226.CrossRefGoogle Scholar
[35]Shelah, S., The spectrum problem I, ℵε-saturatedmodels, the main gap, Israel Journal of Mathematics, vol. 43 (1982), pp. 324–356.CrossRefGoogle Scholar
[36]Shelah, S., The spectrum problem II, totally transcendental theories and infinite depth, Israel Journal of Mathematics, vol. 43 (1982), pp. 357–364.CrossRefGoogle Scholar
[37]Shelah, S., Classification for nonelementary classes I the number of uncountable models of Ψ ϵ Lω1ω, Part A, Israel Journal of Mathematics, vol. 46 (1983), pp. 212–240.CrossRefGoogle Scholar
[38]Shelah, S., Classification for nonelementary classes I the number of uncountable models Ψ ϵ Lω1ω, Part B, Israel Journal of Mathematics, vol. 46 (1983), pp. 241–273.CrossRefGoogle Scholar
[39]Shelah, S., Classification of non elementary classes II, abstract elementary classes, Classification theory (Baldwin, J. T., editor), Lecture Notes in Mathematics, no. 1292, Springer, 1987, pp. 264–419.CrossRefGoogle Scholar
[40]Shelah, S., Universal classes, Classification theory (Baldwin, J. T., editor), Lecture Notes in Mathematics, no. 1292, Springer, 1987, pp. 264–419.CrossRefGoogle Scholar
[41]Shelah, S., Classification theory, Studies in Logic and the Foundations of Mathematics, no. 92, North-Holland, Amsterdam, 1990, 2nd revised edition.Google Scholar
[42]Shelah, S. and Villaveces, A., Categoricity for classes with no maximal models: non forking amalgamation, to appear.Google Scholar
[43]Shelah, S. and Villaveces, A., Towards categoricity for classes with no maximal models, to appear.Google Scholar
[44]Villaveces, A., De Łoś al presente: un teorema aún central en teorí a de la clasificación, Boletin de Matematicas, nueva serie, vol. IV (1997), pp. 1–17.Google Scholar