Definition:Directed Preordering

Definition
Let $\struct {S, \precsim}$ be a preordered set.

Then $\struct {S, \precsim}$ 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 preordering is also known as a filtered (preordered) set or upward directed set.

The term directed set can also be found, but can be confused with a directed ordering.

Also see

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