Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-28T22:37:01.711Z Has data issue: false hasContentIssue false
  • English
  • Français

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:palacin@math.univ-lyonl.fr
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:wagner@math.univ-lyonl.fr
*
Universitat de Barcelona, Departament de Lógica, Història I Filosofia de la Ciència, Montalegre 6, 08001 Barcelona, Spain
Get access Rights & Permissions [Opens in a new window]

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

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.)

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