Axiom:Meet Semilattice Filter Axioms
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Definition
Let $\struct {S, \wedge, \preccurlyeq}$ be a meet semilattice.
Let $F \subseteq S$ be a non-empty subset of $S$.
$F$ is a filter of $S$ if and only if $F$ satisifes the axioms:
\((\text {MSF 1})\) | $:$ | $F$ is an upper section of $S$: | \(\ds \forall x \in F: \forall y \in S:\) | \(\ds x \preccurlyeq y \implies y \in F \) | |||||
\((\text {MSF 2})\) | $:$ | $F$ is a subsemilattice of $S$: | \(\ds \forall x, y \in F:\) | \(\ds x \wedge y \in F \) |
These criteria are called the meet semilattice filter axioms.