More on SOP1 and SOP2

Saharon Shelah; Alexander Usvyatsov

doi:10.1016/j.apal.2008.02.003

More on SOP₁ and SOP₂

Saharon Shelah, Alexander Usvyatsov

Research output: Contribution to journal › Article › peer-review

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 {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 language	English
Pages (from-to)	16-31
Number of pages	16
Journal	Annals of Pure and Applied Logic
Volume	155
Issue number	1
DOIs	https://doi.org/10.1016/j.apal.2008.02.003
State	Published - Aug 2008
Externally published	Yes

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.

Keywords

Keisler ordering
Rank
SOP
SOP

Access to Document

10.1016/j.apal.2008.02.003

Cite this

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title = "More on SOP1 and SOP2",

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. D{\v z}amonja, 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.",

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author = "Saharon Shelah and Alexander Usvyatsov",

note = "Funding Information: This is Publication no. 844 on Shelah{\textquoteright}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.",

year = "2008",

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AU - Usvyatsov, Alexander

N1 - 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.

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N2 - 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.

AB - 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.

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