Axiom:Filter Axioms
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Definition
Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\FF \subseteq S$.
$\FF$ is a filter of $\struct {S, \preccurlyeq}$ if and only if $\FF$ satisfies the following conditions:
\((1)\) | $:$ | \(\ds \FF \ne \O \) | |||||||
\((2)\) | $:$ | \(\ds x, y \in \FF \implies \exists z \in \FF: z \preccurlyeq x, z \preccurlyeq y \) | |||||||
\((3)\) | $:$ | \(\ds \forall x \in \FF: \forall y \in S: x \preccurlyeq y \implies y \in \FF \) |
These criteria are called the filter axioms.