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Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

A filter of $\left({S, \preccurlyeq}\right)$ (or a filter on $\left({S, \preccurlyeq}\right)$) is a subset $\mathcal F \subseteq S$ which satisfies the following conditions:

\((1)\)   $:$   \(\displaystyle \mathcal F \ne \varnothing \)             
\((2)\)   $:$   \(\displaystyle x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y \)             
\((3)\)   $:$   \(\displaystyle \forall x \in \mathcal F: \forall y \in S: x \preccurlyeq y \implies y \in \mathcal F \)             

Proper Filter

Let $\mathcal F$ be a filter on $\left({S, \preccurlyeq}\right)$.

Then $\mathcal F$ is a proper filter on $S$ if and only if $\mathcal F \ne S$.

That is, if and only if $\mathcal F$ is a proper subset of $S$.

Also see

  • Results about filters can be found here.