### Non-forking frames in abstract elementary classes

by Jarden and Shelah. [JrSh:875]

Annals Pure and Applied Logic, 2013

The stability theory of first order theories was initiated by
Saharon Shelah in 1969. The classification of abstract elementary
classes was initiated by Shelah, too. In several papers, he
introduced
non-forming relations. Later, Shelah (2009) [17, 11] introduced
the
good non-forking frame, an axiomatization of the non-forking
notion.
We improve results of Shelah on good non-forming grames, mainly
by
weakening the stability hypothesis in several important theorems,
replacing it by the almost lambda-stability hypothesis: The
number
of types over a model of cardinality lambda is at most
lambda^+ . We present conditions on K_lambda, that imply
the existence of a
model in K_{lambda^{+n}} for all n . We do this by providing
sufficiently strong conditions on K_lambda, that they are inherited
by a properly chosen subclass of K_{lambda^+} . What are these
conditions? We assume that there is a `non-forking' relation
which
satisfies the properties of the non-forking relation on superstable
first order theories. Note that here we deal with models of
fixed
cardinality lambda .
While in Shelah (2009) [17,II] we assume stability in lambda,
so we
can use brimmed (=limit) models, here we assume almost stability
only, but we add an assumption: The conjugation property.
In the context of elementary classes, the superstability assumption
gives the existence of types with well-defined dimension and
the
omega-stability assumption gives the existence and uniqueness
of
models prime over sets. In our context, the local character
assumption
is an analog to superstability and the density of the class of
uniqueness triples with respect to the relation preccurlyeq is
the analog to omega-stability.

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