\begin{definition}[Definition:Prime Filter (Order Theory)]
Let $\left({S, \preceq}\right)$ be an ordered set.
Let $F$ be a filter in $\left({S, \preceq}\right)$.
$F$ is a '''prime filter''' {{iff}}:
:$\forall x, y \in S: \left({x \vee y \in F \implies x \in F \lor y \in F}\right)$
\end{definition}