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Dependent pairs

Published online by Cambridge University Press:  12 March 2014

Ayhan Günaydin  and
Philipp Hieronymi
Ayhan Günaydin
Affiliation:
Centro de Matematica e Aplicacoes Fundamentais, Ay Prof. Gama Pinto, 2, 1649-003, Lisboa, Portugal, E-mail: [email protected]
Philipp Hieronymi
Affiliation:
Centro de Matematica e Aplicacoes Fundamentais, Ay Prof. Gama Pinto, 2, 1649-003, Lisboa, Portugal, E-mail: [email protected]
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Abstract

We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property, regardless of whether it is dense or discrete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 377 - 390
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Berenstein, Alexander, Ealy, Clifton, and Günaydin, Ayhan, Thorn independence in the field of real numbers with a small multiplicative group, Annals of Pure and Applied Logic, vol. 150 (2007), no. 1-3, pp. 1–18.CrossRefGoogle Scholar
[2]van den Dries, Lou, The field of reals with a predicate for the powers of two, Manuscripta Mathematica, vol. 54 (1985), pp. 187–195.CrossRefGoogle Scholar
[3]van den Dries, Lou, T-convexity and tame extensions. II, this Journal, vol. 62 (1997), no. 1, pp. 14–34.Google Scholar
[4]van den Dries, Lou, Dense pairs of o-minimal structures, Fundamenta Mathematical vol. 157 (1998), pp. 61–78.CrossRefGoogle Scholar
[5]van den Dries, Lou and Günaydin, Ayhan, The fields of real and complex numbers with a small multiplicative group, Proceedings of the London Mathematical Society. Third Series, vol. 93 (2006), no. 1, pp. 43–81.CrossRefGoogle Scholar
[6]van den Dries, Lou and Lewenberg, Adam H., T-convexity and tame extensions, this Journal, vol. 60 (1995), no. 1, pp. 74–102.Google Scholar
[7]Evertse, Jan-Hendrik, Schlickewei, Hans Peter, and Schmidt, Wolfgang M., Linear equations in variables which lie in a multiplicative group, Annals of Mathematics. Second Series, vol. 155 (2002), no. 3, pp. 807–836.CrossRefGoogle Scholar
[8]Hieronymi, Philipp, Defining the set of integers in expansions of the real field by a closed discrete set, Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 2163–2168.CrossRefGoogle Scholar
[9]Hieronymi, Philipp, The realfield with an irrational power function and a dense multiplicative subgroup, Journal of the London Mathematical Society (2), vol. 83 (2011), pp. 153–167.CrossRefGoogle Scholar
[10]Macpherson, Dugald, Marker, David, and Steinhorn, Charles, Weakly o-minimal structures and real closed fields, Transactions of the American Mathematical Society, vol. 352 (2000), no. 12, pp. 5435–5483 (electronic).CrossRefGoogle Scholar
[11]Miller, Chris, Tameness in expansions of the realfield. Logic Colloquium '01 (Vienna) (Baaz, Matthias, Krajíček, Jan, and Friedman, Sy D., editors), Lecture Notes in Logic, vol. 20, Association for Symbolic Logic, 2005, pp. 281–316.CrossRefGoogle Scholar
[12]Poizat, Bruno, A course in Model Theory. An introduction to contemporary mathematical logic, Universitext, Springer-Verlag, New York, 2000, Translated from the French by Moses Klein and revised by the author.Google Scholar
[13]Shelah, Saharon, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Annals of Pure and Applied Logic, vol. 3 (1971), no. 3, pp. 271–362.Google Scholar
[14]Tychonievich, Michael A., Defining additive subgroups of the reals from convex subsets, Proceedings of the American Mathematical Society, vol. 137 (2009), no. 10, pp. 3473–3476.CrossRefGoogle Scholar