TY - JOUR

T1 - Non-forking frames in abstract elementary classes

AU - Jarden, Adi

AU - Shelah, Saharon

PY - 2013/3

Y1 - 2013/3

N2 - 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-forking relations. Later, Shelah (2009) [17, II] introduced the good non-forking frame, an axiomatization of the non-forking notion.We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost λ-stability hypothesis: The number of types over a model of cardinality λ is at most λ+. We present conditions on Kλ, that imply the existence of a model in Kλ+n for all n. We do this by providing sufficiently strong conditions on Kλ, that they are inherited by a properly chosen subclass of Kλ+. 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 a fixed cardinality, λ.While in Shelah (2009) [17, II] we assume stability in λ, 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 -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 bs is the analog to stability.

AB - 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-forking relations. Later, Shelah (2009) [17, II] introduced the good non-forking frame, an axiomatization of the non-forking notion.We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost λ-stability hypothesis: The number of types over a model of cardinality λ is at most λ+. We present conditions on Kλ, that imply the existence of a model in Kλ+n for all n. We do this by providing sufficiently strong conditions on Kλ, that they are inherited by a properly chosen subclass of Kλ+. 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 a fixed cardinality, λ.While in Shelah (2009) [17, II] we assume stability in λ, 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 -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 bs is the analog to stability.

KW - AEC

KW - Categoricity

KW - Galois types

KW - Non-forking

KW - Saturated model

KW - Universal

UR - http://www.scopus.com/inward/record.url?scp=84871034745&partnerID=8YFLogxK

U2 - 10.1016/j.apal.2012.09.007

DO - 10.1016/j.apal.2012.09.007

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AN - SCOPUS:84871034745

SN - 0168-0072

VL - 164

SP - 135

EP - 191

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 3

ER -