Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T01:59:09.669Z Has data issue: false hasContentIssue false
  • English
  • Français

Local homogeneity

Published online by Cambridge University Press:  12 March 2014

Bektur Baizhanov  and
John T. Baldwin
Bektur Baizhanov
Affiliation:
Institute for Problems of Informaticsand Control, Pushkin Str. 125, Almaty 480100., Kazakhstan, E-mail: baizhanov@ipic.kz
John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, 851 S. Morgan Street Chicago, IL 60607, USA, E-mail: jbaldwin@math.uic.edu
Get access Rights & Permissions [Opens in a new window]

Abstract.

We study the expansion of stable structures by adding predicates for arbitrary subsets. Generalizing work of Poizat-Bouscaren on the one hand and Baldwin-Benedikt-Casanovas-Ziegler on the other we provide a sufficient condition (Theorem 4.7) for such an expansion to be stable. This generalization weakens the original definitions in two ways: dealing with arbitrary subsets rather than just submodels and removing the ‘small’ or ‘belles paires’ hypothesis. We use this generalization to characterize in terms of pairs, the ‘triviality’ of the geometry on a strongly minimal set (Theorem 2.5). Call a set A benign if any type over A in the expanded language is determined by its restriction to the base language. We characterize the notion of benign as a kind of local homogenity (Theorem 1.7). Answering a question of [8] we characterize the property that M has the finite cover property over A (Theorem 3.9).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 4 , December 2004 , pp. 1243 - 1260
Copyright
Copyright © Association for Symbolic Logic 2004

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]Baizhanov, B., Baldwin, J.T., and Shelah, S., Subsets of superstable structures are weakly benign, submitted.Google Scholar
[2]Baldwin, J. T., Fundamentals of Stability Theory, Springer-Verlag, 1988.CrossRefGoogle Scholar
[3]Baldwin, J. T. and Benedikt, M., Stability theory, permutations of indiscernible and embedded finite models, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 4937–4969.CrossRefGoogle Scholar
[4]Baldwin, J. T. and Holland, K., Constructing ω-stable structures: Rank 2 fields, this Journal, vol. 65 (2000), pp. 371–391.Google Scholar
[5]Baldwin, J. T. and Shelah, S., Second order quantifiers and the complexity of theories, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 229–302.CrossRefGoogle Scholar
[6]Bouscaren, E., Dimensional order property and pairs of models, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 205–231.CrossRefGoogle Scholar
[7]Buechler, S., Pseudoprojective strongly minimal sets are locally projective, this Journal, vol. 56 (1991), pp. 1184–1194.Google Scholar
[8]Casanovas, E. and Ziegler, M., Stable theories with a new predicate, this Journal, vol. 66 (2001), pp. 1127–1140.Google Scholar
[9]Hodges, W... Model theory, Cambridge University Press, 1993.CrossRefGoogle Scholar
[10]Lascar, D., Stability in Model Theory, Longman, 1987, originally published in French as Stabilite en Théorie des Modèles (1986).Google Scholar
[11]Pillay, A., An introduction to stability theory, Clarendon Press, Oxford, 1983.Google Scholar
[12]Poizat, B., Paires de structure stables, this Journal, vol. 48 (1983), pp. 239–249.Google Scholar
[13]Shelah, S., Classification Theory and the Number of Nonisomorphic Models, North-Holland, 1978.Google Scholar