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Abstract

Generalizing Keisler’s notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $\omega _1$ 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 $\mathcal P _\kappa (\lambda )$.