Department of Mathematical Sciences, University of Wisconsin —Milwaukee, Milwaukee, Wisconsin 53201
Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p(M). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl(X) ∩ D, Y = acl(Y) ∩ D and X ∩ Y ≠ ∅,
Theorem 1. Let p ∈ S(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 a ∈ H∖acl(A), b ∈ G∖acl(A) there are a′ ∈ H, b′ ∈ G such that a′ ∈ acl(abb′A)∖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)
1 Research partially supported by NSF grant MCS-8202294