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Ranks and pregeometries in finite diagrams

✍ Scribed by Olivier Lessmann


Book ID
104307496
Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
228 KB
Volume
106
Category
Article
ISSN
0168-0072

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✦ Synopsis


The study of classes of models of a ΓΏnite diagram was initiated by S. Shelah 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 β„΅0-stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is β„΅0-stable. We can also obtain pregeometries with respect to this dependence relation. The dependence 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 concepts are used to generalize many of the classical results for models of a totally transcendental ΓΏrst-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are di erent, as we do not have compactness. 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.


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