Definition:Prime Filter (Order Theory)
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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