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Uniqueness of Limit Models in Classes with Amalgamation

R. Grossberg·M. VanDieren·A. Villaveces
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Abstract

Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the Löwenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M. This theorem extends results of Shelah, Kolman and Shelah, and Shelah and Villaveces. A preliminary version of our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes.

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R. Grossberg, M. VanDieren, A. Villaveces. 2015-02-05. Uniqueness of Limit Models in Classes with Amalgamation. https://arxiv.org/abs/math/0509338

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