Nonregular ideals
Tom 254 / 2021
Streszczenie
Generalizing Keisler’s notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on must be somewhere \omega _1-dense. We prove a dichotomy about degrees of regularity for \kappa -complete ideals on successor cardinals \kappa and apply this to show that Taylor’s Theorem does not generalize to higher cardinals. In particular, the existence of a nonregular ideal on \omega _2 does not imply the existence of an \omega _2-dense ideal on \omega _2. We obtain similar results for normal ideals on \mathcal P _\kappa (\lambda ).