The Journal of Symbolic Logic

Research Article

The geometry of weakly minimal types

Steven Buechler 1

Department of Mathematical Sciences, University of Wisconsin —Milwaukee, Milwaukee, Wisconsin 53201

Abstract

Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let MT be uncountable and saturated, H = p(M). We say DH is locally modular if for all X, YD with X = acl(X) ∩ D, Y = acl(Y) ∩ D and XY ≠ ∅,

Theorem 1. Let pS(A) be weakly minimal and D the realizations of 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 aH∖acl(A), bG∖acl(A) there are a′ ∈ H, b′ ∈ G such that a′ ∈ acl(abbA)∖acl(aA). Similarly when H and G are the realizations of complete types or strong types over A.

(Received October 05 1983)

(Revised April 04 1984)

(Revised October 05 1984)

Footnotes

1   Research partially supported by NSF grant MCS-8202294