directed set

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

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

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 $(A,\le )$ such that whenever $a,b\in A$ there is an $x\in A$ such that $x\le a$ and $x\le b$.

Note: Many authors do not require $\le$ 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.

Title	directed set
Canonical name	DirectedSet
Date of creation	2013-03-22 12:54:00
Last modified on	2013-03-22 12:54:00
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	11
Author	yark (2760)
Entry type	Definition
Classification	msc 06A06
Synonym	upward-directed set
Synonym	upward directed set
Related topic	Cofinality
Related topic	AccumulationPointsAndConvergentSubnets
Defines	residual
Defines	cofinal
Defines	downward-directed set
Defines	downward directed set
Defines	filtered set