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The Comparability Numbers and the Incomparability Numbers

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We introduce new cardinal invariants of a poset, called the comparability number and the incomparability number. We determine their value for well-known posets, such as \(\omega ^\omega \), \(\mathcal {P}(\omega )/\textrm{fin}\), the Turing degrees \(\mathcal {D}\), the quotient algebra \(\textsf {Borel}(2^\omega )/\textsf {null}\), the ideals \(\textsf {meager}\) and \(\textsf {null}\). Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on \(\omega \). We also consider these invariants for ideals on \(\omega \) and on \(\omega _1\).

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The author thanks Yusuke Hayashi for discussing this study with him. The author thanks Hiroshi Sakai, who gave him comments on Section 3. Theorem 3.8 was obtained through private communication with Jorge Antonio Cruz Chapital. He also would like to thank Jörg Brendle, who suggested to him the idea of the proof of Theorem 7.3. In order to prove the results in Section 10, The author was given helpful comments by Dilip Raghavan and Michael Hrušák. The result in Section 12 is due to the private communication with Paul Larson. The author would like to thank the anonymous referees, who gave him numerous helpful comments.


This work was supported by JSPS KAKENHI Grant Number JP22J20021.

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Tatsuya Goto wrote the entire manuscript.

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Correspondence to Tatsuya Goto.

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Goto, T. The Comparability Numbers and the Incomparability Numbers. Order (2024).

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