Definition:Filter/Proper Filter

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

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

That is, $F$ is a proper subset of $S$.

Also see

 * Definition:Filter on Set