Remarks on generalized ultrafilter, dominating and reaping numbers

Shimon Garti, Saharon Shelah Fundamenta Mathematicae MSC: Primary 03E17; Secondary 03E55. DOI: 10.4064/fm595-9-2019 Published online: 6 March 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 $.

Authors

  • Shimon GartiEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Jerusalem 91904, Israel
    e-mail
  • Saharon ShelahEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Jerusalem 91904, Israel and
    Department of Mathematics
    Rutgers University
    New Brunswick, NJ 08854, U.S.A.
    http://www.math.rutgers.edu/~shelah
    e-mail

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