Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 883–898 | Cite as

Katětov and Katětov-Blass orders on \(F_\sigma \)-ideals

  • Hiroaki Minami
  • Hiroshi Sakai


We study the structures \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\), where \(F_\sigma \mathsf {ideals}\) is the family of all \(F_\sigma \)-ideals over \(\omega \), and \(\le _{\mathrm {K}}\) and \(\le _{\mathrm {KB}}\) denote the Katětov and Katětov-Blass orders on ideals. We prove the following:
  • \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\) are upward directed.

  • The least cardinalities of cofinal subfamilies of \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\) are both equal to \({\mathfrak {d}}\). Moreover those of unbounded subfamilies are both equal to \({\mathfrak {b}}\).

  • The family of all summable ideals is unbounded in both \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\).


\(F_\sigma \)-ideals Katětov order Katětov-Blass order 

Mathematics Subject Classification

03E17 03E04 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bell, M.: On the combinatorial principle \(P(c)\). Fundam. Math. 114(2), 149–157 (1981)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Blass, A.: Combinatorial cardinal characteristics of the continuuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, pp. 395–489. Springer, Berlin (2010)CrossRefGoogle Scholar
  3. 3.
    Hernández-Hernández, F., Hrušák, M.: Cardinal invariants of analytic \(P\)-ideals. Can. J. Math. 59(3), 575–595 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hrušák, M.: Combinatorics of filters and ideals. In: Babinkostova, L., Caicedo, A.E., Geschke, S., Scheepers, M. (eds.) Set Theory and Its Applications. Contemporary Mathematics, vol. 533, pp. 29–69. American Mathematical Society, Providence (2011)Google Scholar
  5. 5.
    Laflamme, C.: Zapping small filters. Proc. Am. Math. Soc. 114(2), 535–544 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mazur, K.: \(F_\sigma \)-ideals and \(\omega _1 \omega _1^*\)-gaps in the boolean algebras \(\cal P ( \omega ) / I\). Fundam. Math. 138(2), 103–111 (1991)MathSciNetGoogle Scholar
  7. 7.
    Solecki, S.: Analytic ideals and their applications. Ann. Pure Appl. Log. 99(1–3), 51–72 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Vojtáš, P.: Generalized Galois-Tukey connections between explicit relations on classical objects of real analysis. In: Judah, H. (ed.) Set Theory of the Reals, Israel Mathematical Conference Proceedings, vol. 6, pp. 619–643 (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Aichi Gakuin UniversityNisshinJapan
  2. 2.Kobe UniversityKobeJapan

Personalised recommendations