Definition:Ultrafilter on Set/Definition 4

Definition
Let $S$ be a non-empty set.

Let $\mathcal F$ be a non-empty set of subsets of $S$.

Then $\mathcal F$ is an ultrafilter on $S$ iff both of the following hold:


 * $\mathcal F$ has the finite intersection property.
 * For all $U \subseteq S$, either $U \in \mathcal F$ or $U^c \in \mathcal F$.

where $U^c$ is the complement of $U$ in $S$.

Also see

 * Equivalence of Definitions of Ultrafilter on Set