Abstract
In this paper, we study some new examples of ideals on \(\omega \) with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin–Keisler order.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beros, K.A.: Weak Rudin–Keisler reductions on projective ideals. Fundam. Math. 232(1), 56–78 (2016)
Causey, R.M.: Proximity to \(\ell _p\) and \(c_0\) in Banach spaces. J. Funct. Anal. 269, 3952–4005 (2015)
Fremlin, David H.: The partially ordered sets of measure theory and Tukey’s ordering. Note di Mat. 11, 177–214 (1991)
He, J., Hrušák, M., Rojas-Rebolledo, D., Solecki, S.: Tukey order among \(F_\sigma \) ideals. J. Symb. Log. 86(2), 855–870 (2021)
Isbell, J.R.: Seven cofinal types. J. Lond. Math. Soc. 4(2), 651–654 (1972)
Kechris, A.S.: Classical Descriptive Set Theory, Volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York (1995)
Louveau, A., Velickovic, B.: Analytic ideals and cofinal types. Ann. Pure Appl. Log. 99, 171–195 (1999)
Sakai, H.: On Katětov and Katětov-Blass orders on analytic P-ideals and Borel ideals. Arch. Math. Log. 57, 317–327 (2018)
Solecki, S.: Analytic ideals and their applications. Ann. Pure Appl. Log. 99, 51–72 (1999)
Solecki, S., Todorcevic, S.: Cofinal types of topological directed orders. Ann. l’Institut Fourier (Grenoble) 54(6), 1877–1911 (2005)
Solecki, S., Todorcevic, S.: Avoiding families and Tukey functions on the nowhere-dense ideal. J. Inst. Math. Jussieu 10(2), 405–435 (2011)
Tukey, J.W.: Convergence and Uniformity in Topology, Number 2 in Annals of Mathematics Studies. Princeton University Press, Princeton (1940)
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 research of the second author is supported in part by NSF grants DMS-1201494 and DMS-1764320.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Beros, K.A., Larson, P.B. Maximal Tukey types, P-ideals and the weak Rudin–Keisler order. Arch. Math. Logic 63, 325–352 (2024). https://doi.org/10.1007/s00153-023-00897-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-023-00897-z