\begin{definition}[Definition:Directed Set]
Let $\struct {S, \precsim}$ be a preordered set.
Then $\struct {S, \precsim}$ is a '''directed set''' {{iff}} every pair of elements of $S$ has an upper bound in $S$:
:$\forall x, y \in S: \exists z \in S: x \precsim z$ and $y \precsim z$
\end{definition}