Nonregular ideals

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 $\mathcal P_\kappa(\lambda)$.