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Base matrices of various heights

Part of: Set theory

Published online by Cambridge University Press:  20 April 2023

Jörg Brendle Open the ORCID record for Jörg Brendle [Opens in a new window]
Jörg Brendle*
Affiliation:
Graduate School of System Informatics, Kobe University, Rokko-dai 1-1, Nada-ku, Kobe 657-8501, Japan
*
e-mail: brendle@kobe-u.ac.jp
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Abstract

A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height ${\mathfrak h}$, where ${\mathfrak h}$ is the distributivity number of ${\cal P} (\omega ) / {\mathrm {fin}}$. We show that if the continuum ${\mathfrak c}$ is regular, then there is a base matrix of height ${\mathfrak c}$, and that there are base matrices of any regular uncountable height $\leq {\mathfrak c}$ in the Cohen and random models. This answers questions of Fischer, Koelbing, and Wohofsky.

Keywords

Base matrixrefining matrixdistributivity numbersplitting number

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum
Secondary: 03E05: Other combinatorial set theory 03E35: Consistency and independence results
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 4 , December 2023 , pp. 1237 - 1243
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was partially supported by Grant-in-Aid for Scientific Research (C) 18K03398 from the Japan Society for the Promotion of Science.

References

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