Nonregular ideals

Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $ω_1$ must be somewhere $ω_1$-dense. We prove a dichotomy about degrees of regularity for $κ$-complete ideals on successor cardinals $κ$ and apply this to show that Taylor's Theorem does not generalize to higher cardinals. In particular, the existence of a nonregular ideal on $ω_2$ does not imply the existence of an $ω_2$-dense ideal on $ω_2$. We obtain similar results for normal ideals on $\mathcal P_κ(λ)$.

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