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Saturation and solvability in abstract elementary classes with amalgamation

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Abstract

Theorem 0.1 Let \(\mathbf {K}\) be an abstract elementary class (AEC) with amalgamation and no maximal models. Let \(\lambda > {LS}(\mathbf {K})\). If \(\mathbf {K}\) is categorical in \(\lambda \), then the model of cardinality \(\lambda \) is Galois-saturated.

This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: \(\mathbf {K}\) has a unique limit model in each cardinal below \(\lambda \), (when \(\lambda \) is big-enough) \(\mathbf {K}\) is weakly tame below \(\lambda \), and the thresholds of several existing categoricity transfers can be improved.

We also prove a downward transfer of solvability (a version of superstability introduced by Shelah):

Corollary 0.2 Let \(\mathbf {K}\) be an AEC with amalgamation and no maximal models. Let \(\lambda> \mu > {LS}(\mathbf {K})\). If \(\mathbf {K}\) is solvable in \(\lambda \), then \(\mathbf {K}\) is solvable in \(\mu \).

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  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

    Sebastien Vasey

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  1. Sebastien Vasey
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Vasey, S. Saturation and solvability in abstract elementary classes with amalgamation. Arch. Math. Logic 56, 671–690 (2017). https://doi.org/10.1007/s00153-017-0561-8

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  • DOI: https://doi.org/10.1007/s00153-017-0561-8

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