Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T10:42:40.163Z Has data issue: false hasContentIssue false
  • English
  • Français

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: thyttine@cc.helsinki.fi
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University Jerusalem, Israel Rutgers University, Hill CTR-Bush, New Brunswick, NJ, U.S.A., E-mail: shelah@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

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

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. 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