## You are here

Homedirected set

## Primary tabs

# directed set

A *directed set* is a partially ordered set $(A,\leq)$ such that whenever $a,b\in A$ there is an $x\in A$ such that $a\leq x$ and $b\leq x$.

A subset $B\subseteq A$ is said to be *residual* if there is $a\in A$ such that $b\in B$ whenever $a\leq b$, and *cofinal* if for each $a\in A$ there is $b\in B$ such that $a\leq b$.

A directed set is sometimes called an *upward-directed set*.
We may also define the dual notion:
a *downward-directed set* (or *filtered set*) is a partially ordered set $(A,\leq)$ such that whenever $a,b\in A$ there is an $x\in A$ such that $x\leq a$ and $x\leq b$.

Note: Many authors do not require $\leq$ to be antisymmetric, so that it is only a pre-order (rather than a partial order) with the given property. Also, it is common to require $A$ to be non-empty.

## Mathematics Subject Classification

06A06*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections