Weakly normal filters and irregular ultrafilters

Abstract:

For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafilters, are then proved: Let U be a uniform ultrafilter over a regular cardinal $\kappa$. (a) If $\kappa = {\lambda ^ + }$, then U is not $(\lambda ,{\lambda ^ + })$-regular iff U has a least function f such that $\{ \xi < {\lambda ^ + }|{\text {cf}}(f(\xi )) = \lambda \} \in U$. (b) If $\omega \leqslant \mu < \kappa$ and U is not $(\omega ,\mu )$-regular, then U has a least function.