Definition:Directed Preordering

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

Then $\left({S, \precsim}\right)$ is a directed set 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 $y \precsim z$

Also known as
A directed set is also known as a directed preorder, filtered set or upward directed set.

Also see

 * Definition:Downward Directed Set
 * Definition:Directed Subset
 * Definition:Directed Colimit