Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-31T09:47:25.644Z Has data issue: false hasContentIssue false
  • English
  • Français

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: aberenst@uniandes.edu.co, URL: www.matematicas.uniandes.edu.co/~aberenst
Alf Dolich
Affiliation:
East Stroudsburg University, 200 Prospect St. East Stroudsburg, Pennsylvania 18301, USA, E-mail: adolich@po-box.esu.edu, URL: www.dolich.com
Alf Onshuus
Affiliation:
Universidad de Los Andes, CRA 1 NO 18A-10, Bogotá, Colombia, E-mail: aonshuus@uniandes.edu.co
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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