(\(\kappa ,\theta \))-weak normality (Q431613)

An ultrafilter \(D\) on a cardinal \(\kappa\) is weakly normal if for every regressive function \(f\) on \(\kappa\) there is some \(\alpha_* < \kappa\) such that \(\{ i<\kappa : f(i) \leq \alpha_* \} \in D\).NEWLINENEWLINE The authors deal with the notion of weak normality. Let \(\bar{\lambda} = \langle \lambda_i: i<\kappa \rangle\) be a sequence of cardinals with limit \(\lambda\). They characterize the situation of \(|\prod_{i<\kappa} \lambda_i/D| = \lambda\). Further on, they find a necessary condition for a positive answer to a question of Monk on the depth of Boolean algebras.