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Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 317–327 | Cite as

On Katětov and Katětov–Blass orders on analytic P-ideals and Borel ideals

  • Hiroshi Sakai
Article
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Abstract

Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \(F_\sigma \) ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:
  • The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.

  • The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders.

In the course of the proof of the latter result, we also prove that for any analytic ideal \(\mathcal {I}\) there is a Borel ideal \(\mathcal {J}\) with \(\mathcal {I} \subseteq \mathcal {J}\).

Keywords

Katětov order Analytic P-ideal Borel ideal 

Mathematics Subject Classification

03E04 03E15 54H05 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Kobe UniversityKobeJapan

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