Locally finite weakly minimal theories

Abstract

Suppose T is a weakly minimal theory and p a strong 1-type having locally finite but nontrivial geometry. That is, for any M [boxvR] T and finite F⊆ p( M), there is a finite G⊆ p( M) such that acl( F)∩ p( M)  ∪ gϵ G acl({ g})∩ p M; however, we cannot always choose G  F. Then there are formulas θ( x) and E( x, y) so that θϵ p and for any M[boxvR] T, E defines an equivalence relation with finite classes on θ( M)/ E definably inherits the structure of either a projective or affine space over some finite field. We then specify what other structure θ( M)/ E may inherit: there is some collection of definable subspaces of finite codimension and some set of algebraic points, which in the affine case may be in the canonically associated vector space, Up to acl(∅), no further structure is possible. If we assume T is weakly minimal and has a strong type p as above, and also that T is unidimensional, we obtain a global description of any model of T in terms of those structures mentioned in the previous paragraph.