8,861 research outputs found
Limit Models in Strictly Stable Abstract Elementary Classes
In this paper, we examine the locality condition for non-splitting and
determine the level of uniqueness of limit models that can be recovered in some
stable, but not superstable, abstract elementary classes. In particular we
prove:
Suppose that is an abstract elementary class satisfying
1. the joint embedding and amalgamation properties with no maximal model of
cardinality .
2. stabilty in .
3. .
4. continuity for non--splitting (i.e. if and is a
limit model witnessed by for some limit
ordinal and there exists so that does
not -split over for all , then does not -split over
).
For and limit ordinals both with cofinality , if satisfies symmetry for non--splitting (or just
-symmetry), then, for any and that are
and -limit models over , respectively, we have that
and are isomorphic over .Comment: This article generalizes some results from arXiv:1507.0199
Excellent Abstract Elementary Classes are tame
The assumption that an AEC is tame is a powerful assumption permitting
development of stability theory for AECs with the amalgamation property. Lately
several upward categoricity theorems were discovered where tameness replaces
strong set-theoretic assumptions.
We present in this article two sufficient conditions for tameness, both in
form of strong amalgamation properties that occur in nature. One of them was
used recently to prove that several Hrushovski classes are tame.
This is done by introducing the property of weak -uniqueness which
makes sense for all AECs (unlike Shelah's original property) and derive it from
the assumption that weak (\LS(\K),n)-uniqueness, (\LS(\K),n)-symmetry and
(\LS(\K),n)-existence properties hold for all . The conjunction of
these three properties we call \emph{excellence}, unlike \cite{Sh 87b} we do
not require the very strong (\LS(\K),n)-uniqueness, nor we assume that the
members of \K are atomic models of a countable first order theory. We also
work in a more general context than Shelah's good frames.Comment: 26 page
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