Let $\mathbb{F}_1$ and $\mathbb{F}_2$ be two filters on the same set. The following terminology is commonly used to describe the relation of $\mathbb{F}_1$ to $\mathbb{F}_2$

$\mathbb{F}_2$ is said to be finer than $\mathbb{F}_1$ if $\mathbb{F}_1 \subseteq \mathbb{F}_2$

$\mathbb{F}_2$ is said to be coarser than $\mathbb{F}_1$ if $\mathbb{F}_1 \supseteq \mathbb{F}_2$

$\mathbb{F}_2$ is said to be strictly finer than $\mathbb{F}_1$ if $\mathbb{F}_1 \subset \mathbb{F}_2$

$\mathbb{F}_2$ is said to be strictly coarser than $\mathbb{F}_1$ if $\mathbb{F}_1 \supset \mathbb{F}_2$

$\mathbb{F}_1$ and $\mathbb{F}_2$ are said to be comparable if either $\mathbb{F}_1 \subseteq \mathbb{F}_2$ or $\mathbb{F}_1 \supseteq \mathbb{F}_2$