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

TL;DR

This paper determines the exact rank of certain filter limits when the filter is Borel of rank 1, and explores how ranks of these limits can vary, including cases where ranks drop to 1.

Contribution

It provides exact calculations of filter ranks for Borel filters of rank 1 and estimates ranks of $$-limits, revealing they can decrease to 1 regardless of initial ranks.

Findings

Exact rank of $$-Fubini sum for Borel filters of rank 1

Ranks of $$-limits can be as low as 1

Filters of countable type and their ranks are analyzed

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.