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EQUIVALENT DEFINITIONS OF SUPERSTABILITY IN TAME ABSTRACT ELEMENTARY CLASSES

Published online by Cambridge University Press:  09 January 2018

RAMI GROSSBERG  and
SEBASTIEN VASEY
RAMI GROSSBERG
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA, USAE-mail:rami@cmu.eduURL: http://math.cmu.edu/∼rami
SEBASTIEN VASEY
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA, USAE-mail:sebv@cmu.eduURL: http://math.cmu.edu/∼svasey/
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Abstract

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:

  • Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:

    1. (1) For all high-enough λ,Khas no long splitting chains.

    2. (2) For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.

    3. (3) For all high-enough λ,Khas a unique limit model of cardinality λ.

    4. (4) For all high-enough λ,Khas a superlimit model of cardinality λ.

    5. (5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.

    6. (6) There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.

This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Keywords

Primary 03C48Secondary 03C4503C5203C5503C7503E55abstract elementary classessuperstabilitytamenessindependenceclassification theorysuperlimitsaturatedsolvabilitygood framelimit model
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1387 - 1408
Copyright
Copyright © The Association for Symbolic Logic 2017 

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