Home Mathematics Chains in the Rudin-Frolík order
Article
Licensed
Unlicensed Requires Authentication

Chains in the Rudin-Frolík order

  • Joanna Jureczko EMAIL logo
Published/Copyright: December 6, 2024
Become an author with De Gruyter Brill

Abstract

The aim of this paper is to construct chains of length (2κ)+ in the Rudin-Frolík order of β κ for every infinite κ.

Acknowledgement

The author is very grateful to the anonymous reviewer for their insight in reading the previous version of this paper. Their remarks undoubtedly avoided many inaccuracies and made the text more readable.

  1. Communicated by L’ubica Holá

References

[1] Baker, J.—Kunen, K.: Limits in the uniform ultrafilters, Trans. Amer. Math. Soc. 353(10) (2001), 4083–4093.10.1090/S0002-9947-01-02843-4Search in Google Scholar

[2] Booth, D.: Ultrafilters on a countable set, Ann. Math. Logic 2(1) (1970/71), 1–24.10.1016/0003-4843(70)90005-7Search in Google Scholar

[3] Bukovský, L.—Butkovičová, E.: Ultrafilters with ℵ0 predecessors in Rudin-Frolík order, Comment. Math. Univ. Carolin. 22(3) (1981), 429–447.Search in Google Scholar

[4] Butkovičová, E.: Ultrafilters without immediate predecessors in Rudin-Frolík order, Comment. Math. Univ. Carolin. 23(4) (1982), 757–766.Search in Google Scholar

[5] Butkovičová, E.: Long chains in Rudin-Frolík order, Comment. Math. Univ. Carolin. 24(3) (1983), 563–570.Search in Google Scholar

[6] Butkovičová, E.: Subsets of βwithout an infimum in Rudin-Frolík order, Proc. of the 11th Winter School on Abstract Analysis (Železná Ruda, 1983), Rend. Circ. Mat. Palermo (2) (1984), Suppl. no. 3, 75–80.Search in Google Scholar

[7] Butkovičová, E.: Decreasing chains without lower bounds in the Rudin-Frolík order, Proc. Amer. Math. Soc. 109(1) (1990), 251–259.10.1090/S0002-9939-1990-1007490-8Search in Google Scholar

[8] Comfort, W. W.—Negrepontis, S.: The Theory of Ultrafilters, Springer 1974.10.1007/978-3-642-65780-1Search in Google Scholar

[9] Frolík, Z.: Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87–91.10.1090/S0002-9904-1967-11653-7Search in Google Scholar

[10] Gitik, M.: Some constructions of ultrafilters over a measurable cardinal, Ann. Pure Appl. Logic 171(8) (2020), Art. 102821.10.1016/j.apal.2020.102821Search in Google Scholar

[11] Hart, K. P.: Long chains in the Rudin-Frolík order for uncountable cardinals, https://arxiv.org/pdf/2303.14252 (2024).Search in Google Scholar

[12] Jech, T.: Set Theory. 3rd millennium edition, revised and expanded. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Search in Google Scholar

[13] Jureczko, J.: Ultrafilters without immediate predecessors in Rudin-Frolík order for regulars, Results Math. 77 (2022), Art. 230.10.1007/s00025-022-01762-wSearch in Google Scholar

[14] Jureczko, J.: Chains in the Rudin-Frolík order for regulars, Raporty Katedry Telekomunikacji i Teleinformatyki, 2022; dona.pwr.edu.plSearch in Google Scholar

[15] Jureczko, J.: A note on special subsets of the Rudin-Frolík orderfor regulars, Math. Slovaca 73(4) (2023), 825–834.10.1515/ms-2023-0060Search in Google Scholar

[16] Jureczko, J.: Decreasing chains without lower bounds in the Rudin Frolík order for regulars, https://arxiv.org/pdf/2304.01398.pdfSearch in Google Scholar

[17] Jureczko, J.: On some constructions of ultrafilters over a measurable cardinal, in preparation.Search in Google Scholar

[18] Kanamori, A.: Ultrafilters over a measurable cardinal, Ann. Math. Logic 11 (1976), 315–356.10.1016/0003-4843(76)90012-7Search in Google Scholar

[19] Kunen, K.: Set Theory, London, UK: College Publications, 2011.Search in Google Scholar

[20] Kunen, K.: Weak P-points in* Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), pp. 741–749, Colloq. Math. Soc. Jnos Bolyai, 23, North-Holland, Amsterdam-New York, 1980.Search in Google Scholar

[21] Rudin, M. E.: Types of Ultrafilters. In: Topology Seminar Wisconsin 1965, Princeton Universiy Press, 1966.10.1515/9781400882076-021Search in Google Scholar

[22] Rudin, M. E.: Partial orders on the types in βℕ, Trans. Amer. Math. Soc. 155 (1971), 353–362.10.2307/1995690Search in Google Scholar

Received: 2023-06-11
Accepted: 2024-08-09
Published Online: 2024-12-06
Published in Print: 2024-12-15

© 2024 Mathematical Institute Slovak Academy of Sciences

Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0112/pdf
Scroll to top button