\begin{definition}[Definition:Filter/Proper Filter]
Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\FF$ be a filter on $\struct {S, \preccurlyeq}$.
Then:
:$\FF$ is a '''proper filter''' on $S$
{{iff}}:
:$\FF \ne S$
That is, {{iff}} $\FF$ is a proper subset of $S$.
\end{definition}