# Sh:875

- Jarden, A., & Shelah, S. (2013).
*Non-forking frames in abstract elementary classes*. Ann. Pure Appl. Logic,**164**(3), 135–191. arXiv: 0901.0852 DOI: 10.1016/j.apal.2012.09.007 MR: 3001542 -
Abstract:

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. - published version (57p)

Bib entry

@article{Sh:875, author = {Jarden, Adi and Shelah, Saharon}, title = {{Non-forking frames in abstract elementary classes}}, journal = {Ann. Pure Appl. Logic}, fjournal = {Annals of Pure and Applied Logic}, volume = {164}, number = {3}, year = {2013}, pages = {135--191}, issn = {0168-0072}, mrnumber = {3001542}, mrclass = {03C48 (03C35 03C45 03C52)}, doi = {10.1016/j.apal.2012.09.007}, note = {\href{https://arxiv.org/abs/0901.0852}{arXiv: 0901.0852}}, arxiv_number = {0901.0852} }