Abstract
Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \(F_\sigma \) ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:
-
The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.
-
The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders.
In the course of the proof of the latter result, we also prove that for any analytic ideal \(\mathcal {I}\) there is a Borel ideal \(\mathcal {J}\) with \(\mathcal {I} \subseteq \mathcal {J}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bartoszyński, T., Judah, H.: Set Theory: On the Structure of the Real Line. A K Peters Ltd, Wellesley, MA (1995)
Hrušák, M.: Combinatorics of Filters and Ideals, Set Theory and Its Applications, Contemporary Mathematics, vol. 533, pp. 29–69. American Mathematical Society, Providence, RI (2011)
Hrušák, M., Meza Alcántara, D.: Universal submeasures and ideals. Questions Answers Gen. Topology 31(2), 65–69 (2013)
Hrušák, M., Meza Alcántara, D., Minami, H.: Pair-splitting, pair-reaping and cardinal invariants of \(F_\sigma \)-ideals. J. Symb. Logic 75(2), 661–677 (2010)
Katětov, M.: Products of filters. Comment. Math. Univ. Carol. 9, 173–189 (1968)
Katětov, M.: On Descriptive Classes of Functions, Theory of Sets and Topology (in Honour of Felix Hausdorff, 1868–1942), pp. 265–278. VEB Deutsch, Verlag Wissensch, Berlin (1972)
Mathias, A.R.D.: A Remark on Rare Filters, Infinite and Finite Sets (Colloq., Keszthely, 1973; Dedicated to P. Erdös on His 60th Birthday), Colloquia Mathematica Societatis Janos Bolyai, vol. 10, pp. 1095–1097. North-Holland, Amsterdam (1975)
Mazur, K.: \(F_\sigma \)-ideals and \(\omega _1 \omega _1^*\)-gaps in \(\cal{P} ( \omega ) / I\). Fund. Math. 138(2), 103–111 (1991)
Meza Alcántara, D.: Ideals and Filters on Countable Sets, Ph.D. thesis, UNAM México (2009)
Minami, H., Sakai, H.: Katětov and Katětov–Blass orders on \(F_\sigma \)-ideals. Arch. Math. Logic 55(7–8), 883–898 (2016)
Solecki, S.: Analytic ideals and their applications. Ann. Pure Appl. Logic 99(1–3), 51–72 (1999)
Talagrand, M.: Compacts de fonctions mesurables et filtres non mesurables. Stud. Math. 67(1), 13–43 (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sakai, H. On Katětov and Katětov–Blass orders on analytic P-ideals and Borel ideals. Arch. Math. Logic 57, 317–327 (2018). https://doi.org/10.1007/s00153-017-0572-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-017-0572-5