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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: [email protected]
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: [email protected]
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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

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References

REFERENCES

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