December 2022 The higher Cichoń diagram in the degenerate case
Jörg Brendle
Tsukuba J. Math. 46(2): 255-269 (December 2022). DOI: 10.21099/tkbjm/20224602255

Abstract

For a regular uncountable cardinal $\kappa$, we discuss the order relationship between the unbounding and dominating numbers $\mathfrak{b}_{\kappa}$ and $\mathfrak{d}_{\kappa}$ on $\kappa$ and cardinal invariants of the higher meager ideal $\mathscr{M}_{\kappa}$. In particular, we obtain an almost complete characterization of $\mathsf{add}(\mathscr{M}_{\kappa})$ and $\mathsf{cof}(\mathscr{M}_{\kappa})$ in terms of $\mathsf{cov}(\mathscr{M}_{\kappa})$ and $\mathsf{non}(\mathscr{M}_{\kappa})$ and unbounding and dominating numbers, and we provide models showing that there are no restrictions on the value of $\mathsf{non}(\mathscr{M}_{\kappa})$ in the degenerate case $2^{\lt \kappa} > \kappa$ except $2^{\lt \kappa} \le \mathsf{non}(\mathscr{M}_{\kappa}) \le 2^{\kappa}$. The corresponding question for $\mathsf{cof}(\mathscr{M}_{\kappa})$ remains open. Our results answer questions of joint work of the author with Brooke-Taylor, Friedman, and Montoya [BBFM, Questions 29 and 32].

Citation

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Jörg Brendle. "The higher Cichoń diagram in the degenerate case." Tsukuba J. Math. 46 (2) 255 - 269, December 2022. https://doi.org/10.21099/tkbjm/20224602255

Information

Published: December 2022
First available in Project Euclid: 17 March 2023

MathSciNet: MR4561580
Digital Object Identifier: 10.21099/tkbjm/20224602255

Subjects:
Primary: 03E05
Secondary: 03E17 , 03E35

Keywords: dominating number , Forcing , higher cardinal invariants , higher meager ideal , unbounding number

Rights: Copyright © 2022 University of Tsukuba, Institute of Mathematics

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Vol.46 • No. 2 • December 2022
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