Definition:Ultrafilter on Set/Definition 3

Definition
Let $S$ be a set.

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

Then $\mathcal F$ is an ultrafilter (on $S$) :
 * for every $A \subseteq S$, either $A \in \mathcal F$ or $\complement_S \left({A}\right) \in \mathcal F$

where $\complement_S \left({A}\right)$ is the relative complement of $A$ in $S$, that is, $S \setminus A$.

Also see

 * Equivalence of Definitions of Ultrafilter on Set