\begin{definition}[Definition:Filtered Subset]
Let $\left({S, \precsim}\right)$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is a '''filtered subset''' of $S$ {{iff}}:
:$\forall x, y \in H: \exists z \in H: z \precsim x$ and $z \precsim y$
\end{definition}