The geometry of weakly minimal types

Journal of Symbolic Logic 50 (4):1044-1053 (1985)
Abstract
Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let $M \models T$ be uncountable and saturated, H = p(M). We say $D \subset H$ is locally modular if for all $X, Y \subset D$ with $X = \operatorname{acl}(X) \cap D, Y = \operatorname{acl}(Y) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of $\operatorname{stp}(a/A)$ for some a realizing p. Then D is locally modular or p has Morley rank 1. Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all $a \in H\backslash\operatorname{acl}(A), b \in G\operatorname{acl}(A)$ there are a' ∈ H, b' ∈ G such that $a' \in \operatorname{acl}(abb' A)\backslash\operatorname{acl}(aA)$ . Similarly when H and G are the realizations of complete types or strong types over A
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DOI 10.2307/2273989
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Unidimensional Theories Are Superstable.Ehud Hrushovski - 1990 - Annals of Pure and Applied Logic 50 (2):117-137.
Vaught's Conjecture for Superstable Theories of Finite Rank.Steven Buechler - 2008 - Annals of Pure and Applied Logic 155 (3):135-172.
Meager Forking.Ludomir Newelski - 1994 - Annals of Pure and Applied Logic 70 (2):141-175.
A Dichotomy Theorem for Regular Types.Ehud Hrushovski & Saharon Shelah - 1989 - Annals of Pure and Applied Logic 45 (2):157-169.

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