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Ample thoughts

Published online by Cambridge University Press:  12 March 2014

Daniel Palacín  and
Frank O. Wagner
Daniel Palacín*
Affiliation:
Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan UMR5208, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France, E-mail:[email protected]
Frank O. Wagner
Affiliation:
Université de Lyon CNRS, Université Lyon 1, Institut Camille Jordan Umr5208, 43 Bd Du 11 Novembre 1918, 69622 Villeurbanne Cedex, France, E-mail:[email protected]
*
Universitat de Barcelona, Departament de Lógica, Història I Filosofia de la Ciència, Montalegre 6, 08001 Barcelona, Spain
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Abstract

Non-n-ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

Keywords

stablesimpleone-basedCM-trivialn-ampleinternalanalysableclosurelevelflatultraflatcanonical base property
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 489 - 510
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1]Buechler, Steven, Vaught's conjecture for superstable theories of finite rank, Annals of Pure and Applied Logic, vol. 155 (2008), pp. 135–172.CrossRefGoogle Scholar
[2]Buechler, Steven and Hoover, Colleen, The classification of small types of rank ω I, this Journal, vol. 66 (2001), pp. 1884–1898.Google Scholar
[3]Campana, Frédéric, Algébricité et compacité dans l'espace des cycles d'un espace analytique complexe, Mathematische Annaien, vol. 251 (1980), pp. 7–18.CrossRefGoogle Scholar
[4]Chatzidakis, ZoÉ, A note on canonical bases and modular types in supersimple theories, Confluentes Mathematici, vol. 4 (2012), no. 3.CrossRefGoogle Scholar
[5]Evans, David, Ample dividing, this Journal, vol. 68 (2003), pp. 1385–1402.Google Scholar
[6]Fujiki, Akira, On the Douady space of a compact complex space in the category , Nagoya Mathematical Journal, vol. 85 (1982), pp. 189–211.CrossRefGoogle Scholar
[7]Hall, Peter, Some sufficient conditions for a group to be nilpotent, Illinois Journal of Mathematics, vol. 2 (1958), pp. 787–801.CrossRefGoogle Scholar
[8]Hall, Peter and Hartley, Brian, The stability group of a series of subgroups, Proceedings of the London Mathematical Society. Third Series, vol. 16 (1966), pp. 1–39.CrossRefGoogle Scholar
[9]Hrushovski, Ehud, Locally modular regular types, Classification Theory (Baldwin, John, editor), Springer-Verlag, Berlin, 1985.Google Scholar
[10]Hrushovski, Ehud, Contributions to stable model theory, Ph.D. thesis, 1986.Google Scholar
[11]Hrushovski, Ehud, A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
[12]Hrushovski, Ehud, The Manin-Mumford conjecture and the model theory of difference fields. Annals of Pure and Applied Logic, vol. 112 (2001), no. 1, pp. 43–115.CrossRefGoogle Scholar
[13]Hrushovski, Ehud, Palacin, Daniel, and Pillay, Anand, On the canonical base property, preprint, 2012.Google Scholar
[14]Juhlin, Prerna Bihani, Fine stucture of dependence in superstable theories of finite rank, Ph.D. thesis, University of Notre Dame, Indiana, 2010.Google Scholar
[15]Kaloujnine, Leo, Über gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen, Bericht über die Mathematiker-Tagung in Berlin, Januar 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953, pp. 164–172.Google Scholar
[16]Kowalski, Piotr and Pillay, Anand, Quantifier elimination for algebraic D-groups, Transactions of the American Mathematical Society, vol. 358 (2005), pp. 167–181.CrossRefGoogle Scholar
[17]Houcine, Abderezak Ould and Tent, Katrin, Ampleness in the free group, preprint, 2012.Google Scholar
[18]Pillay, Anand, The geometry of forking and groups of finite Morley rank, this Journal, vol. 60 (1995), pp. 1251–1259.Google Scholar
[19]Pillay, Anand, Geometric stability theory, Oxford Logic Guides, vol. 32, Oxford University Press, Oxford, 1996.CrossRefGoogle Scholar
[20]Pillay, Anand, A note on CM-triviality and the geometry of forking, this Journal, vol. 65 (2000), pp. 474–480.Google Scholar
[21]Pillay, Anand, Notes on analysability and canonical bases, e-print available at http://vrew.math.uiuc.edu/People/pillay/remark.zoe.pdf, 2001.Google Scholar
[22]Pillay, Anand, Model-theoretic consequences of a theorem of Campana and Fujiki, Fundamenta Mathematicae, vol. 174 (2002), no. 2, pp. 187–192.CrossRefGoogle Scholar
[23]Pillay, Anand and Ziegler, Martin, Jet spaces of varieties over differential and difference fields, Selecta Mathematica. New Series, vol. 9 (2003), pp. 579–599.CrossRefGoogle Scholar
[24]Wagner, Frank O. (editor), Stable groups, LMS lecture Note Series, vol. 240, Cambridge University Press, Cambridge, 1997.CrossRefGoogle Scholar
[25]Wagner, Frank O. (editor), Simple theories, Mathematics and Its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar
[26]Wagner, Frank O. (editor), Some remarks on one-basedness, this Journal, vol. 69 (2004), pp. 34–38.Google Scholar