The geometry of weakly minimal types

Journal of Symbolic Logic 50 (4):1044-1053 (1985)
  Copy   BIBTEX

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

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,221

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Dp-Minimality: Basic Facts and Examples.Alfred Dolich, John Goodrick & David Lippel - 2011 - Notre Dame Journal of Formal Logic 52 (3):267-288.
The classification of small weakly minimal sets. II.Steven Buechler - 1988 - Journal of Symbolic Logic 53 (2):625-635.
Isolated types in a weakly minimal set.Steven Buechler - 1987 - Journal of Symbolic Logic 52 (2):543-547.
Weakly o-minimal structures and some of their properties.B. Sh Kulpeshov - 1998 - Journal of Symbolic Logic 63 (4):1511-1528.
Les automorphismes d'un ensemble fortement minimal.Daniel Lascar - 1992 - Journal of Symbolic Logic 57 (1):238-251.
Pseudoprojective strongly minimal sets are locally projective.Steven Buechler - 1991 - Journal of Symbolic Logic 56 (4):1184-1194.
One theorem of Zil′ber's on strongly minimal sets.Steven Buechler - 1985 - Journal of Symbolic Logic 50 (4):1054-1061.

Analytics

Added to PP
2009-01-28

Downloads
17 (#738,848)

6 months
1 (#1,028,709)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Unidimensional theories are superstable.Ehud Hrushovski - 1990 - Annals of Pure and Applied Logic 50 (2):117-137.
Locally modular theories of finite rank.Steven Buechler - 1986 - Annals of Pure and Applied Logic 30 (1):83-94.
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.

View all 13 citations / Add more citations

References found in this work

ℵ0-Categorical, ℵ0-stable structures.G. Cherlin, L. Harrington & A. H. Lachlan - 1985 - Annals of Pure and Applied Logic 28 (2):103-135.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.

Add more references