Definition:Prime Filter (Order Theory)

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

Let $F$ be a filter in $\left({S, \preceq}\right)$.

$F$ is a prime filter :
 * $\forall x, y \in S: \left({x \vee y \in F \implies x \in F \lor y \in F}\right)$