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Preprint Number 916

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916. Monica M. VanDieren and Sebastien Vasey
On the structure of categorical abstract elementary classes with amalgamation
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Submission date: 4 September 2015.

Abstract:

For K an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah.

Theorem
Let μ > LS (K). If K is categorical in a λ > ℶ_{(2^{μ})^+}, then:
1) Whenever M_0, M_1, M_2 in K_μ are such that M_1 and M_2 are limit over M_0, we have M_1 ≅_{M_0} M_2.
2) If μ > LS (K), the model of size λ is μ-saturated.
3) If μ > ℶ_{(2^{LS (K)})^+} and λ > ℶ_{(2^{μ^+})^+}, then there exists a type-full good μ-frame with underlying class the saturated models in K_μ. Our main tool is the symmetry property of splitting (previously isolated by the first author). The key lemma deduces symmetry from failure of the order property.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases:

Full text arXiv 1509.01488: pdf, ps.


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