# Definition:Filter

## Definition

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.