Ranks and pregeometries in finite diagrams

The study of classes of models of a finite diagram was initiated by S. Shel ah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the N-0-stable case which share many of the formal properties of forking. This is achieved by considering a rank for th is framework which is bounded when the diagram D is N-0-stable. We can also obtain pregeometries with respect to this dependence relation. The depende nce relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these c oncepts are used to generalize many of the classical results for models of a totally transcendental first-order theory, in fact, strong analogies aris e: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have co mpactness. This is illustrated with positive results (categoricity) as well as negative results (construction of nonisomorphic models). We also give a proof of a Two Cardinal Theorem for this context. (C) 2000 Elsevier Scienc e B.V. All rights reserved. MSC: 03C45; 03C75.