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Some remarks on generic structures

Published online by Cambridge University Press:  12 March 2014

David M. Evans  and
Mark Wing Ho Wong
David M. Evans
Affiliation:
University of East Anglia, School of Mathematics, Norwich, Nr4 7Tj, UK, E-mail: [email protected]
Mark Wing Ho Wong
Affiliation:
University of East Anglia, School of Mathematics, Norwich, Nr4 7Tj, UK, E-mail: [email protected]
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Abstract

We show that the ℵ0-categorical structures produced by Hrushovski's predimension construction with a control function fit neatly into Shelah's SOPn hierarchy: if they are not simple, then they have SOP3 and NSOP4. We also show that structures produced without using a control function can be undecidable and have SOP.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1143 - 1154
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Baldwin, John T. and Shelah, Saharon, Randomness and semigenericity, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 1359–1376.CrossRefGoogle Scholar
[2]Evans, David M., ℵ0-categorical structures with apredimension, Annals of Pure and Applied Logic, vol. 116 (2002), pp. 157–186.CrossRefGoogle Scholar
[3]Evans, David M. and Pantano, M. E., ℵ0-categorical structures with arbitrarily fast growth of algebraic closure, this Journal, vol. 67 (2002), pp. 897–909.Google Scholar
[4]Hodges, Wilfrid, Model Theory, Cambridge University Press, 1993.CrossRefGoogle Scholar
[5]Hrushovski, Ehud, A stable ℵ0-categorical pseudoplane, 1988, unpublished notes.Google Scholar
[6]Hrushovski, Ehud, A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
[7]Hrushovski, Ehud, Simplicity and the Lascar group, 1997, unpublished notes.Google Scholar
[8]Kueker, D. W. and Laskowski, M. C., On generic structures, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 175–183.CrossRefGoogle Scholar
[9]Pourmahdian, Massoud, Smooth classes without AC and Robinson theories, this Journal, vol. 67 (2002), pp. 1274–1294.Google Scholar
[10]Pourmahdian, Massoud, Simple generic structures, Annals of Pure and Applied Logic, vol. 121 (2003), pp. 227–260.CrossRefGoogle Scholar
[11]Shelah, Saharon, Toward classifying unstable theories, Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229–255.CrossRefGoogle Scholar
[12]Shelah, Saharon and Spencer, Joel, Zero-one laws for sparse random graphs. Journal of the American Mathematical Society, vol. 1 (1988), pp. 97–115.CrossRefGoogle Scholar
[13]Wagner, Frank O., Simple Theories, Kluwer, Dordrecht, 2000.CrossRefGoogle Scholar
[14]Wing, MarkWong, Ho, Ph.D. thesis, University of East Anglia, Norwich, 06 2007.Google Scholar