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 ${\mathcal{P}}_{\kappa }(\lambda )$.