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:

$(1): \quad \mathcal F \ne \varnothing$

$(2): \quad x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y$

$(3): \quad \forall x \in \mathcal F: \forall y \in S: x \preccurlyeq y \implies y \in \mathcal F$

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

• Results about filters can be found here.