Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T08:37:48.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Cardinal Invariants of Analytic $P$-Ideals

Published online by Cambridge University Press:  20 November 2018

Fernando Hernández-Hernández  and
Michael Hrušák
Fernando Hernández-Hernández
Affiliation:
Instituto de Matemáticas, UNAM, Unidad Morelia, Morelia, Michoacán, México C.P. 58089 email: [email protected], [email protected]
Michael Hrušák
Affiliation:
Instituto de Matemáticas, UNAM, Unidad Morelia, Morelia, Michoacán, México C.P. 58089 email: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the cardinal invariants of analytic $P$-ideals, concentrating on the ideal $Z$ of asymptotic density zero. Among other results we prove $\min \,\{\mathfrak{b},\,\operatorname{cov}(\mathcal{N})\,\}\,\le \,\operatorname{cov}*\,(Z)\,\le \,\max \{\mathfrak{b},\,\text{non(}\mathcal{N}\text{)}\}$.

Keywords

03E1703E40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 3 , 01 June 2007 , pp. 575 - 595
Copyright
Copyright © Canadian Mathematical Society 2007

References

[Ba] Bartoszyński, T., Invariants of Measure and Category (1999). Available at arXiv:math.LO/9910015.Google Scholar
[BJ] Bartoszyński, T. and Judah, H., Set theory. On the structure of the real line. A K Peters, Wellesley, MA, 1995.Google Scholar
[BD] Baumgartner, J. E. and Dordal, P., Adjoining dominating functions. J. Symbolic Logic, 50(1985), no. 1, 94–101.Google Scholar
[Br] Brendle, J., Mob families and mad families. Arch. Math. Logic 37(1997), no. 3, 183–197.Google Scholar
[BS] Brendle, J. and Shelah, S., Ultrafilters on ω—their ideals and their cardinal characteristics. Trans. Amer. Math. Soc. 351(1999), no. 7, 2643–2674.Google Scholar
[BY] Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families. Ann. Pure Appl. Logic 132(2005), no. 2-3, 271–312.Google Scholar
[Fa1] Farah, I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Mem. Amer. Math. Soc. 148(2000), no. 702.Google Scholar
[Fa2] Farah, I., Analytic Hausdorff gaps. In: Set Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 58, American Mathematical Society, Providence, RI, 2002, pp. 65–72.Google Scholar
[Fa3] Farah, I., How many Boolean algebra. P(ℕ)/I are there? Illinois J. Math. 46(2002), no. 4, 999–1033.Google Scholar
[Fa4] Farah, I., Luzin gaps. Trans. Amer. Math. Soc. 356(2004), no. 6, 2197–2239.(electronic).Google Scholar
[Fa5] Farah, Ilijas, Analytic Hausdorff gaps. II. The density zero ideal. Israel J. Math. 154(2006), 235–246.Google Scholar
[Fr1] Fremlin, D. H., The partially ordered sets of measure theory and Tukey's ordering. Note Mat. 11(1991), 177–214.Google Scholar
[Fr2] Fremlin, David H., Measure theory. Set theoretic measure theory. Torres Fremlin, Colchester, England, 2004. Available at http://www/essex.ac.uk/maths/staff/fremlin/my.htm. Google Scholar
[HZ1] Hrušák, M. and Zapletal, J., Forcing with quotients. To appear in Arch. Math. Logic.Google Scholar
[Je] Jech, T., Set theory. Second edition. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1997.Google Scholar
[Ku] Kunen, K., Set Theory. An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics 102, North-Holland, Amsterdam, 1980.Google Scholar
[La] Laflamme, C., Zapping small filters. Proc. Amer. Math. Soc. 114(1992), no. 2, 535–544.Google Scholar
[LV] Louveau, A. and Veličković, B., Analytic ideals and cofinal types. Ann. Pure Appl. Logic 99(1999), no. 1-3, 171–195.Google Scholar
[Sh] Shelah, S., Proper and Improper Forcing. Second edition. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1998.Google Scholar
[So] Solecki, S., Analytic ideals and their applications. Ann. Pure Appl. Logic 99(1999), no. 1-3, 51–72.Google Scholar
[To] Todorčević, S., Analytic gaps. Fund. Math. 150(1996), no. 1, 55–66.Google Scholar
[Va] Vaughan, J. E., Small uncountable cardinals and topology.With an appendix by S. Shelah. In: Open Problems in Topology. North-Holland, Amsterdam, 1990, pp. 195–218.Google Scholar