Definition:Ultrafilter on Set/Definition 1

Definition
Let $S$ be a set.

Let $\mathcal F \subseteq \powerset S$ be a filter on $S$.

Then $\mathcal F$ is an ultrafilter (on $S$) :
 * there is no filter on $S$ which is strictly finer than $\mathcal F$

or equivalently, :
 * 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