Remarks on generalized ultrafilter, dominating and reaping numbers
Volume 250 / 2020
Abstract
The following statements are the main results of the paper:
(a) $\operatorname{cf} (\mathfrak {u}) \gt \omega $ and $\operatorname{cf} (\mathfrak {u}_\kappa ) \gt \omega $ for every uncountable cardinal $\kappa $ where $\mathfrak {u}_\kappa $ is the generalized ultrafilter number.
(b) If $\kappa \gt \aleph _0$ is regular and $\mathfrak {r}_\kappa \lt \mathfrak {d}_\kappa $ then $\mathfrak {r}_\kappa =\mathfrak {u}_\kappa $, where $\mathfrak {r}_\kappa $ is the generalized reaping number and $\mathfrak {d}_\kappa $ is the generalized dominating number.
(c) The relations $\mathfrak {r}_\lambda \lt \mathfrak {d}_\lambda $ and $\mathfrak {u}_\lambda \lt \mathfrak {d}_\lambda $ are consistent for a strong limit singular cardinal $\lambda $.
