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\(1\)-based theories – the main gap for \(a\)-models. (English) Zbl 0849.03022

For a 1-based theory \(T\) with NDOP, any \(a\)-\(k_p(T)\)-saturated model is proved to be prime (in the category of \(a\)-\(k_p(T)\)-saturated models) over a non-forking tree of elements, where the tree height is at most \(|T|^+\) (usually called a structure theorem). For a shallow theory of this kind, the number of \(a\)-\(k_p(T)\)-saturated models of \(T\) of cardinality \(\aleph_a\) is less than \({\mathcal J}_{(2^{|T|})^+} (|\omega + \alpha|)\). Otherwise \(T\) has \(2^\lambda\) of \(a\)-\(k_p(T)\)-saturated models of cardinality \(\lambda\), for most \(\lambda\).

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C50 Models with special properties (saturated, rigid, etc.)
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