# Definition:Directed Preordering

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## Definition

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

Then $\struct {S, \precsim}$ is a **directed preordering** 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

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