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Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 883–898 | Cite as

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

  • Hiroaki Minami
  • Hiroshi Sakai
Article

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 \(( F_\sigma \mathsf {ideals} , \le _{\mathrm {KB}} )\).

Keywords

\(F_\sigma \)-ideals Katětov order Katětov-Blass order 

Mathematics Subject Classification

03E17 03E04 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Aichi Gakuin UniversityNisshinJapan
  2. 2.Kobe UniversityKobeJapan

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