Definition:Ultrafilter on Set

Definition
Let $X$ be a set and $\mathcal F \subset \mathcal P \left({X}\right)$ be a filter on $X$.

Then $\mathcal F$ is called an ultrafilter (on $X$) if it is not contained in a larger filter (on $X$).

That is:
 * if it has no superfilter which is strictly finer than it, or
 * if, whenever $\mathcal G$ is a filter on $X$ and $\mathcal F \subseteq \mathcal G$ holds, then $\mathcal F = \mathcal G$.