Definition:Directed Preordering

Definition

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

Then $\struct {S, \precsim}$ is a directed set if and only if 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.

• Results about directed preorderings can be found here.

Also see

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