Adi Jarden, Saharon Shelah. Non-forking Frames in Abstract Elementary Classes

Natural Sciences / Mathematics / Algebra

Submitted on: Oct 10, 2012, 18:44:15

Description: The stability theory of first order theories was initiated by Saharon Shelah in 1969. The classification of abstract elementary classes was initiated by Shelah, too. In several papers, he introduced non-forking relations. Later, in cite{shh}.II, he introduced the good non-forking frame, an axiomatization of the non-forking notion. We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost $lambda$-stability hypothesis: The number of types over a model of cardinality $lambda$ is at most $lambda^+$. In the context of elementary classes, the superstability assumption gives the existence of types with well-defined dimension and the $omega$-stability assumption gives the existence and uniqueness of models prime over sets. In our context, the local character assumption is an analog to superstability and the density of the class of uniqueness triples with respect to the relation $preceq_{bs}$ is the analog to $omega$-stability.

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