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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 $ A$ to be non-empty.



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

Other names:  upward-directed set, upward directed set
Also defines:  residual, cofinal, downward-directed set, downward directed set, filtered set
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Cross-references: partial order, pre-order, antisymmetric, subset, partially ordered set
There are 31 references to this entry.

This is version 8 of directed set, born on 2002-08-01, modified 2007-02-04.
Object id is 3249, canonical name is DirectedSet.
Accessed 5888 times total.

Classification:
AMS MSC06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)

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