Abstract
We show that there is a \( \varvec{\Sigma }_4^0\) ideal such that it’s neither extendable to any \( \varvec{\Pi }_3^0\) ideal nor above the ideal \( \textrm{Fin}\times \textrm{Fin} \) in the sense of Katětov order, answering a question from M. Hrušák.
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We are grateful to the anonymous referee(s) for their patience and valuable comments including pointing out unproper writings, indicating incomplete proofs and so on, which helps us making the paper more cleaner and readable.
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Jialiang He and Jintao Luo are supported by Science and Technology Department of Sichuan Province (project 2022ZYD0012 and 2023NSFSC1285). Shuguo Zhang is supported by NSFC.
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He, J., Luo, J. & Zhang, S. On the extendability to \(\mathbf {\Pi }_3^0\) ideals and Katětov order. Arch. Math. Logic (2024). https://doi.org/10.1007/s00153-024-00912-x
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DOI: https://doi.org/10.1007/s00153-024-00912-x