Ranks of $\mathcal{F}$-limits of filter sequences

Abstract

We give an exact value of the rank of an $\mathcal{F}$-Fubini sum of filters for the case where $\mathcal{F}$ is a Borel filter of rank $1$. We also consider $\mathcal{F}$-limits of filters ${\mathcal{F}}_i$, which are of the form ${\lim }_{\mathcal{F}}{\mathcal{F}}_i=\left\{A\subset X:\left\{i\in I:A\in {\mathcal{F}}_i\right\}\in \mathcal{F}\right\}$. We estimate the ranks of such filters; in particular we prove that they can fall to $1$ for $\mathcal{F}$ as well as for ${\mathcal{F}}_i$ of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.