directed set (original) (raw)

A subset B⊆A is said to be residual if there is a∈A such that b∈B whenever a≤b, and cofinal if for each a∈A there is b∈B such that a≤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,≤) such that whenever a,b∈A there is an x∈A such that x≤a and x≤b.

Note: Many authors do not require ≤ 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.