Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T21:44:18.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructing ω-stable structures: rank 2 fields

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin  and
Kitty Holland
John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, 851 S. Morgan St., Chicago, IL 60607, USA, E-mail: jbaldwin@uic.edu
Kitty Holland
Affiliation:
Department of Mathematics, Northern Illinois University, Dekalb, IL 60115, USA, E-mail: kholland@math.niu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from ‘primitive extensions’ to the natural numbers a theory Tμ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 371 - 391
Copyright
Copyright © Association for Symbolic Logic 2000

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]Aref'ev, Roman, Baldwin, J. T., and Mazzucco, M., δ-invariant amalgamation classes, to appear in this Journal.Google Scholar
[2]Baldwin, J. T., An almost strongly minimal non-desarguesian projective plane, Transactions of the American Mathematical Society, vol. 342 (1994), pp. 695–711.CrossRefGoogle Scholar
[3]Baldwin, J. T., and Holland, K., Constructing Ω-stable structures: Infinite rank, in preparation.Google Scholar
[4]Baldwin, J. T. and Shelah, S., Randomness and semigenericity, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 1359–1376.CrossRefGoogle Scholar
[5]Baldwin, J. T. and Shi, Niandong, Stable generic structures, Annals of Pure and Applied Logic, vol. 79 (1996), pp. 1–35.CrossRefGoogle Scholar
[6]Baldwin, John T., Problems on ‘pathological structures’, Proceedings of 10th Easter conference in model theory, Wendisches Rietz, April 12–19, 1993 (Weese, Helmut Wolter Martin, editor), 1993, pp. 1–9.Google Scholar
[7]Baudisch, A. and Anand, Pillay, A free pseudospace, preprint.Google Scholar
[8]Berline, C. and Lascar, D., Groupes superstates, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 1–45.CrossRefGoogle Scholar
[9]Borovik, A. and Nesin, A., Groups of finite Morley rank, Oxford University Press, 1994.CrossRefGoogle Scholar
[10]Cherlin, G. L. and Shelah, S., Superstable groups and rings, Annals of Pure and Applied Logic, vol. 18 (1980), pp. 227–270.Google Scholar
[11]Goode, J., Hrushovski's geometries, Proceedings of 7th Easter conference on model theory (Dahn, Helmut Wolter Bernd, editor), 1989, pp. 106–118.Google Scholar
[12]Holland, Kitty, Model completeness of the new strongly minimal sets, to appear in this Journal.Google Scholar
[13]Holland, Kitty, An introduction to the fusion of strongly minimal sets: The geometry of fusions, Archive for Mathematical Logic, vol. 6 (1995), pp. 395–413.CrossRefGoogle Scholar
[14]Holland, Kitty, Strongly minimal fusions of vector spaces, Annals of Pure and Applied Logic, vol. 83 (1997), pp. 1–22.CrossRefGoogle Scholar
[15]Hrushovski, E., Simplicity and the Lascar group, preprint.Google Scholar
[16]Hrushovski, E., Strongly minimal expansions of algebraically closed fields, Israel Journal of Mathematics, vol. 79 (1992), pp. 129–151.CrossRefGoogle Scholar
[17]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
[18]Pillay, A., The geometry of forking and groups of finite Morley rank, this Journal, vol. 60 (1995), pp.1251–1259.Google Scholar
[19]Poizat, Bruno, Le carrÉ de l'egalitÉ, to appear in this Journal.Google Scholar