Definition:Ultrafilter on Set/Definition 4

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

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

Then $\FF$ is an ultrafilter on $S$ both of the following hold:


 * $\FF$ has the finite intersection property
 * For all $U \subseteq S$, either $U \in \FF$ or $U^\complement \in \FF$

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

Also see

 * Equivalence of Definitions of Ultrafilter on Set