# Nonregular ideals

## TL;DR

This paper investigates degrees of regularity for ideals on successor cardinals, demonstrating that Taylor's theorem for nonregular ideals on  does not extend to higher cardinals like , and explores related properties of normal ideals.

## Contribution

It establishes a dichotomy for degrees of regularity of - and -complete ideals and shows the limitations of Taylor's theorem at higher cardinals.

## Key findings

- A nonregular ideal on  does not necessarily imply an -dense ideal.
- Taylor's theorem for regularity does not generalize to higher cardinals.
- Similar results hold for normal ideals on ppa(mbda).

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