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One theorem of Zil′ber's on strongly minimal sets

Published online by Cambridge University Press:  12 March 2014

Steven Buechler
Steven Buechler*
Affiliation:
Department of Mathematics, University of Wisconsin—Milwaukee, Milwaukee, Wisconsin 53201
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Abstract

Suppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,

We prove the following theorems.

Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.

(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].

Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 4 , December 1985 , pp. 1054 - 1061
Copyright
Copyright © Association for Symbolic Logic 1985

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References

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