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Katětov and Katětov-Blass orders on \(F_\sigma \)-ideals

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Abstract

We study the structures \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\), where \(F_\sigma \mathsf {ideals}\) is the family of all \(F_\sigma \)-ideals over \(\omega \), and \(\le _{\mathrm {K}}\) and \(\le _{\mathrm {KB}}\) denote the Katětov and Katětov-Blass orders on ideals. We prove the following:

  • \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\) are upward directed.

  • The least cardinalities of cofinal subfamilies of \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\) are both equal to \({\mathfrak {d}}\). Moreover those of unbounded subfamilies are both equal to \({\mathfrak {b}}\).

  • The family of all summable ideals is unbounded in both \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {K}} )\) and \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\).

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Correspondence to Hiroshi Sakai.

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This work was supported by JSPS KAKENHI Grant Number 15K04984.

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Minami, H., Sakai, H. Katětov and Katětov-Blass orders on \(F_\sigma \)-ideals. Arch. Math. Logic 55, 883–898 (2016). https://doi.org/10.1007/s00153-016-0500-0

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