Submission date: 8 July 2015.

Abstract:

We prove:

Main Theorem: Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality μ. Let μ be a cardinal above the the Löwenheim-Skolem number of the class. If K is μ-Galois-stable, has no μ-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two (μ,σ_l)-limits over M, for l in {1,2}, are isomorphic over M.

This theorem extends results of Shelah from [Sh394], [Sh576], [Sh600], Kolman and Shelah in [KoSh] and Shelah and Villaveces from [ShVi]. A preliminary version of our uniqueness theorem, which was circulated in 2006, was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes in [GrVa2]. Preprints of this paper have also influenced the Ph.D. theses of Drueck [Dr] and Zambrano [Za]. This paper also serves the expository role of presenting together the arguments in [Va1] and [Va2] in a more natural context in which the amalgamation property holds and this work provides an approach to the uniqueness of limit models that does not rely on Ehrenfeucht-Mostowski constructions.

Mathematics Subject Classification:

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Full text arXiv 1507.02118: pdf, ps.