Definition:Directed Subset

Definition

Let $\left({S, \precsim}\right)$ be a preordered set.

Let $H$ be a non-empty subset of $S$.

Then $H$ is a directed subset of $S$ if and only if:

$\forall x, y \in H: \exists z \in H: x \precsim z$ and $y \precsim z$