# 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 if and only if:
$\forall x, y \in S: \left({x \vee y \in F \implies x \in F \lor y \in F}\right)$