Pseudoprojective strongly minimal sets are locally projective

Journal of Symbolic Logic 56 (4):1184-1194 (1991)

Abstract
Let D be a strongly minimal set in the language L, and $D' \supset D$ 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 $k < \omega$ such that, for all a, b ∈ D and closed $X \subset D, a \in \mathrm{cl}(Xb) \Rightarrow$ there is a $Y \subset X$ 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 $\S4$ ) 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
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DOI 10.2307/2275467
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References found in this work BETA

Superstable Fields and Groups.G. Cherlin & S. Shelah - 1980 - Annals of Mathematical Logic 18 (3):227-270.
Locally Modular Theories of Finite Rank.Steven Buechler - 1986 - Annals of Pure and Applied Logic 30 (1):83-94.
Intersections of Algebraically Closed Fields.C. J. Ash & John W. Rosenthal - 1986 - Annals of Pure and Applied Logic 30 (2):103-119.
Locally Finite Weakly Minimal Theories.James Loveys - 1991 - Annals of Pure and Applied Logic 55 (2):153-203.

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