PlanetMath: directed set

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

directed set

(Definition)

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 to be non-empty.

"directed set" is owned by yark. [ full author list (2) | owner history (1) ]

(view preamble)