Submission date: 13 February 2015.

Abstract:

In this paper we address a problem posed by Shelah in 1999 to find a suitable notion for superstability for abstract elementary classes in which limit models of cardinality μ are saturated.
Theorem 1. Suppose that K is a χ-tame abstract elementary class with no maximal models satisfying the joint embedding property and the amalgamation property. Suppose μ is a cardinal with μ\geq\beth_{(2^{LS(K)+χ})^+}. Let M be a model of cardinality μ. If K is both χ-stable and μ-stable and satisfies the μ-superstability assumptions, then any two μ-limit models over M are isomorphic over M.
Moreover, we identify sufficient conditions for superlimit models of cardinality μ to exist, for model homogeneous models to be superlimit, and for a union of saturated models to be saturated.

Mathematics Subject Classification: 03C95

Keywords and phrases:

Full text arXiv 1502.04144: pdf, ps.