Definition:Ultrafilter on Set/Definition 1

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

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

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

Also see

 * Equivalence of Definitions of Ultrafilter on Set