Definition:Filter

Definition
Let $$\left({S, \preccurlyeq}\right)$$ be a poset.

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:


 * $$\mathcal F \ne \varnothing$$


 * $$x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y$$


 * $$\forall x \in \mathcal F: \forall y \in S: x \preccurlyeq y \implies y \in \mathcal F$$

Proper Filter
Let $$\mathcal F$$ be a filter $$\mathcal F$$ on a poset $$\left({S, \preccurlyeq}\right)$$ such that $$\mathcal F \ne S$$.

That is, such that $$\mathcal F$$ is a proper subset of $$S$$.

Then $$\mathcal F$$ is a proper filter of $$S$$.

Also see

 * Filter on a Set