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 if and only if:

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

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_7:def 2