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 the following condition holds:

If $$\mathcal G$$ is a filter on $$X$$ and $$\mathcal F \subseteq \mathcal G$$ holds then $$\mathcal F = \mathcal G$$.