More on SOP 1 and SOP 2

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 NSOP 1 theories and give an example of a theory which is NSOP 1 but not simple. We also investigate the connection between maximality in the ordering ◃ ∗ among complete first order theories and the (N)SOP 2 property. We prove that ◃ ∗ -maximality implies SOP 2 and obtain certain results in the other direction. The paper provides a step toward the classification of unstable theories without the strict order property.