Cofinal Types below ℵω

R. O.Y. Shalev

doi:10.1017/jsl.2023.32

Cofinal Types below ℵ_ω

R. O.Y. Shalev

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality <![CDATA[ $\leq \aleph _n$ ]]> is at least <![CDATA[ $c_{n+2}$ ]]>, the <![CDATA[ $(n+2)$ ]]> -Catalan number. Moreover, the class <![CDATA[ $\mathcal D_{\aleph _n}$ ]]> of directed sets of cardinality <![CDATA[ $\leq \aleph _n$ ]]> contains an isomorphic copy of the poset of Dyck <![CDATA[ $(n+2)$ ]]> -paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

Original language	English
Journal	Journal of Symbolic Logic
DOIs	https://doi.org/10.1017/jsl.2023.32
State	Accepted/In press - 2023

Bibliographical note

Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic.

Keywords

03E04
2020 Mathematics Subject Classification

Access to Document

10.1017/jsl.2023.32

Cite this

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abstract = "It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality <![CDATA[ $\leq \aleph _n$ ]]> is at least <![CDATA[ $c_{n+2}$ ]]>, the <![CDATA[ $(n+2)$ ]]> -Catalan number. Moreover, the class <![CDATA[ $\mathcal D_{\aleph _n}$ ]]> of directed sets of cardinality <![CDATA[ $\leq \aleph _n$ ]]> contains an isomorphic copy of the poset of Dyck <![CDATA[ $(n+2)$ ]]> -paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.",

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