Definition:Ultrafilter on Set/Definition 1

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Let $S$ be a set.

Let $\FF \subseteq \powerset S$ be a filter on $S$.

Then $\FF$ is an ultrafilter (on $S$) if and only if:

there is no filter on $S$ which is strictly finer than $\FF$

or equivalently, if and only if:

whenever $\GG$ is a filter on $S$ and $\FF \subseteq \GG$ holds, then $\FF = \GG$.

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