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## Superstability in simple finitary AECs

### Tom 195 / 2007

Fundamenta Mathematicae 195 (2007), 221-268 MSC: Primary 03C45; Secondary 03C52, 03C95. DOI: 10.4064/fm195-3-3

#### Streszczenie

We continue the study of {finitary abstract elementary classes} beyond $\aleph_0$-stability. We suggest a possible notion of superstability for simple finitary AECs, and derive from this notion several good properties for independence. We also study constructible models and the behaviour of Galois types and {weak Lascar strong types} in this context. We show that superstability is implied by {a-categoricity} in a suitable cardinal. As an application we prove the following theorem: Assume that $(\mathbb{K},\preccurlyeq_\mathbb{K})$ is a simple, tame, finitary AEC, a-categorical in some cardinal $\kappa$ above the Hanf number such that $\mathop{\rm cf}\nolimits(\kappa)>\omega$. Then $(\mathbb{K},\preccurlyeq_\mathbb{K})$ is a-categorical in each cardinal above the Hanf number.

#### Autorzy

• Tapani HyttinenDepartment of Mathematics and Statistics
University of Helsinki
P.O. Box 68
FI-00014 Helsinki, Finland
e-mail
• Meeri KesäläDepartment of Mathematics and Statistics
University of Helsinki
P.O. Box 68
FI-00014 Helsinki, Finland
e-mail

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