Definition:Coarser Filter on Set/Strictly Coarser

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Definition

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\FF, \FF' \subset \powerset S$ be two filters on $S$.


Let $\FF \subset \FF'$, that is, $\FF \subseteq \FF'$ but $\FF \ne \FF'$.

Then $\FF'$ is strictly coarser than $\FF$.


Also known as

If $\FF$ is a strictly coarser filter than $\FF'$, then $\FF'$ can also be referred to as a proper superfilter of $\FF$.

However, this is not encouraged, as there exists the danger of confusing this with the concept of a proper filter.


Also see


Sources