More on SOP1 and SOP2

Saharon Shelah, Alexander Usvyatsov

Research output: Contribution to journalArticlepeer-review


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 {white left-pointing small triangle}*-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 {white left-pointing small triangle}* among complete first order theories and the (N)SOP2 property. We prove that {white left-pointing small triangle}*-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.

Original languageEnglish
Pages (from-to)16-31
Number of pages16
JournalAnnals of Pure and Applied Logic
Issue number1
StatePublished - Aug 2008
Externally publishedYes

Bibliographical note

Funding Information:
This is Publication no. 844 on Shelah’s list of publications. The authors would like to thank the Israel Science Foundation for partial support of this research (Grant no. 242/03). The second author warmly thanks Mirna Dzˇamonja for fruitful discussions and support. The authors thank Shani Ben-David for kindly typing parts of this paper.


  • Keisler ordering
  • Rank
  • SOP
  • SOP


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