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The independence property in generalized dense pairs of structures

Published online by Cambridge University Press:  12 March 2014

Alexander Berenstein ,
Alf Dolich  and
Alf Onshuus
Alexander Berenstein
Affiliation:
Universidad de Los Andes, CRA 1 NO 18A-10, Bogotá, Colombia, E-mail: [email protected], URL: www.matematicas.uniandes.edu.co/~aberenst
Alf Dolich
Affiliation:
East Stroudsburg University, 200 Prospect St. East Stroudsburg, Pennsylvania 18301, USA, E-mail: [email protected], URL: www.dolich.com
Alf Onshuus
Affiliation:
Universidad de Los Andes, CRA 1 NO 18A-10, Bogotá, Colombia, E-mail: [email protected]
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Abstract

We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.

Keywords

Rosy theorieslovely pairsdense pairsdependent theoriesstrongly dependent theories
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 391 - 404
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Adler, H., A geometric introduction to forking and thorn-forking, preprint, available at www.amsta.leeds.ac.uk/-adler.Google Scholar
[2]Baizhanov, B. and Baldwin, J., Local homogeneity, this Journal, vol. 69 (2004), pp. 1243–1260.Google Scholar
[3]Yaacov, I. Ben, Pillay, A., and Vassiliev, E., Lovely pairs of models, Annals of Pure and Applied Logic, vol. 122 (2003), pp. 235–261.CrossRefGoogle Scholar
[4]Berenstein, A. and Vassiliev, E., On lovely pairs of geometric structures, Annals of Pure and Applied Logic, vol. 161 (2010), pp. 866–878.CrossRefGoogle Scholar
[5]Boxall, G., NIP for some pair-like theories, Modnet preprint #181.Google Scholar
[6]Buechler, S., Pseudoprojective strongly minimal sets are locally projective, this Journal, vol. 56 (1991), pp. 1184–1194.Google Scholar
[7]Casanovas, E. and Ziegler, M., Stable theories with a new predicate, this Journal, vol. 66 (2001), pp. 1127–1140.Google Scholar
[8]Chang, C.C. and Keisler, H.J., Model theory, third ed., Studies in Logic and Foundations of Mathematics, North Holland, 1990.Google Scholar
[9]Chatzidakis, Z. and Pillay, A., Generic structures and simple theories, Annals of Pure and Applied Logic, (1998), pp. 71–92.Google Scholar
[10]Denef, J. and van den Dries, L., P-adic and real subanalytic sets, The Annals of Mathematics, vol. 128 (1988), pp. 79–138.CrossRefGoogle Scholar
[11]Dolich, A. and Lippel, D.Goodrick, J., Dp-minimality: basic facts and examples, Modnet preprint #207.Google Scholar
[12]Dolich, A., Miller, C., and Steinhorn, C., Structures having o-minimal open core, Transactions of the American Mathematical Society, vol. 362 (2010), pp. 1371–1411.CrossRefGoogle Scholar
[13]van den Dries, L., Dense pairs of o-minimal structures, Fundamenta Mathematicae, vol. 157 (1998), pp. 61–78.CrossRefGoogle Scholar
[14]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Notes Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
[15]Dries, L. van den and Lewenberg, A., T-convexity and tame extensions, this Journal, vol. 60 (1995), pp. 74–102.Google Scholar
[16]Günaydin, A. and Hieronymi, P., Dependent pairs, to be published in this Journal.Google Scholar
[17]Haskell, D. and Macpherson, D., A version ofo-minimalityfor the p-adics, this Journal, vol. 62 (1997), pp. 1075–1092.Google Scholar
[18]Macpherson, D., Marker, D., and Steinhorn, C., Weakly o-minimal structures and real closed fields, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 5435–5483.CrossRefGoogle Scholar
[19]Matthews, L., Cell decomposition and dimension functions in first-order topological structures, Proceedings of the London Mathematical Society, vol. s3-70 (1995), pp. 1–32.CrossRefGoogle Scholar
[20]Onshuus, A., Properties and consequences of thorn-independence, this Journal, vol. 71 (2006), pp. 1–21.Google Scholar
[21]Pillay, A., On externally definable sets and a theorem of Shelah, preprint, available at www.amsta.leeds.ac.uk/~pillay/.Google Scholar
[22]Poizat, B., Paires de structures stables, this Journal, vol. 48 (1983), pp. 239–249.Google Scholar
[23]Shelah, S., Strongly dependent theories, SH863, available at shelah. logic.at/.Google Scholar
[24]Shelah, S., Dependent first order theories, continued, Israel Journal of Mathematics, vol. 173 (2009), no. 1, pp. 1–60.CrossRefGoogle Scholar
[25]Vassiliev, E., Generic pairs of SU-rank 1 structures, Annals of Pure and Applied Logic, vol. 120 (2003), pp. 103–149.CrossRefGoogle Scholar