Definition:Superfilter

Definition
Let $$X$$ be a set, and $$\mathcal P \left({X}\right)$$ be the power set of $$X$$.

Let $$\mathcal F, \mathcal F' \subset \mathcal P \left({X}\right)$$ be two filters on $$X$$.

Then $$\mathcal F'$$ is called a superfilter of $$\mathcal F$$ if $$\mathcal F \subseteq \mathcal F'$$.

Finer / Coarser
If $$\mathcal F'$$ is a superfilter of $$\mathcal F$$, then:


 * $$\mathcal F'$$ is finer than $$\mathcal F$$;
 * $$\mathcal F$$ is coarser than $$\mathcal F'$$.

If $$\mathcal F \subset \mathcal F'$$, i.e. $$\mathcal F \ne \mathcal F'$$, then:


 * $$\mathcal F'$$ is strictly finer than $$\mathcal F$$;
 * $$\mathcal F$$ is strictly coarser than $$\mathcal F'$$;

If $$\mathcal F \subset \mathcal F'$$, then it is possible to refer to $$\mathcal F'$$ as a proper superfilter of $$\mathcal F$$, but this is not advised as there exists the danger of confusing this with the concept of a proper filter.

Comparable Filters
Two filters $$\mathcal F \subset \mathcal F'$$ on a set $$X$$ are comparable iff one is finer than the other.