Definition:Ultrafilter (Order Theory)

Definition
Let $O = \left({S, \preceq}\right)$ be an ordered set.

Let $F$ be a filter in $O$.

Then $F$ is ultrafilter (on $O$)
 * $F$ is proper subset of $S$ and
 * for all filter $G$ in $O$: $\left({F \subseteq G \implies F = G \lor G = S}\right)$