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Pseudoprojective strongly minimal sets are locally projective

Published online by Cambridge University Press:  12 March 2014

Steven Buechler
Steven Buechler*
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
*
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
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Abstract

Let D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We prove

Theorem A. If D is not locally modular, then T′ has Morley rank ω.

We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we prove

Theorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.

The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.

Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 4 , December 1991 , pp. 1184 - 1194
Copyright
Copyright © Association for Symbolic Logic 1991

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