More on SOP1 and SOP2

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Abstract

This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 (1996) 229–255] and [M. Džamonja, S. Shelah, On -maximality, Annals of Pure and Applied Logic 125 (2004) 119–158]. We present a rank function for NSOP1 theories and give an example of a theory which is NSOP1 but not simple. We also investigate the connection between maximality in the ordering among complete first order theories and the (N)SOP2 property. We prove that -maximality implies SOP2 and obtain certain results in the other direction. The paper provides a step toward the classification of unstable theories without the strict order property.

MSC

03C45
03C52
03C95

Keywords

SOP1
SOP2
Rank
Keisler ordering

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