Abstract
We continue the study of Rosenthal families initiated by Damian Sobota. We show that every Rosenthal filter is the intersection of a finite family of ultrafilters that are pairwise incomparable in the Rudin-Keisler partial ordering of ultrafilters. We introduce a property of filters, called an \(r\)-filter, properly between a selective filter and a \(p\)-filter. We prove that every \(r\)-ultrafilter is a Rosenthal family. We prove that it is consistent with ZFC to have uncountably many \(r\)-ultrafilters such that any intersection of finitely many of them is a Rosenthal filter.
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 (1995)
Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Springer, New York (1974)
Erdős, P., Hajnal, A., Máté, A., Rado, R.: Combinatorial Set Theory: Partitions Relations for Cardinals. Akadémiai Kiadó, Budapest (1984)
Grigorieff, S.: Combinatorics on ideals and forcing. Ann. Math. Logic 3, 363–394 (1971)
Jech, T.: Set Theory. Revised and Expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, The Third Millennium Edition (2003)
Komjáth, P., Totik, V.: Problems and Theorems in Classical Set Theory. Springer, Berlin (2006)
Koszmider, P., Martínez-Celis, A.: Rosenthal families, pavings, and generic cardinal invariants. Proc. Am. Math. Soc. 149, 1289–1303 (2021)
Laflamme, C.: Strong meager properties for filters. Fund. Math. 146(3), 283–293 (1995)
Plewik, Sz.: Intersections and unions of ultrafilters without the Baire property, Bull. Polish Acad. Sci. Math. 35, no. 11-12, 805–808 (1987)
Repický, M.: Properties of measure and category in generalized Cohen’s and Silver’s forcing. Acta Univ. Carol., Math. Phys 28(2), 101–115 (1987)
Repický, M.: Cardinal invariants and the collapse of the continuum by Sacks forcing. J. Symb. Logic 73(2), 711–727 (2008)
Rosenthal, H.P.: On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37, 13–36 (1970)
Sobota, D.: Cardinal invariants of the continuum and convergence of measures on compact spaces. Institute of Mathematics, Polish Academy of Sciences (2016). (Ph.D. thesis)
Sobota, D.: Families of sets related to Rosenthal’s lemma. Arch. Math. Logic 58, 53–69 (2019)
Talagrand, M.: Compacts de fonctions mesurables et filtres non mesurables, Studia Mathematica 67, no. l, 13–43 (1980)
Zdomskyy, L.: A semifilter approach to selection principles. Comment. Math. Univ. Carol. 46(3), 525–539 (2005)
Acknowledgements
We would like to thank the anonymous reviewer whose comments and suggestions helped improve and clarify the publication.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author was supported by grant VEGA 2/0097/20.
Rights and permissions
About this article
Cite this article
Repický, M. Rosenthal families, filters, and semifilters. Arch. Math. Logic 61, 131–153 (2022). https://doi.org/10.1007/s00153-021-00779-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-021-00779-2