Axiom:Filter Axioms

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.