Directed Set - Directed Subsets

Directed Subsets

The order relation in a directed sets is not required to be antisymmetric, and therefore directed sets are not always partial orders. However, the term directed set is also used frequently in the context of posets. In this setting, a subset A of a partially ordered set (P,≤) is called a directed subset if it is a directed set according to the same partial order: in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on the elements of A is inherited from P; for this reason, reflexivity and transitivity need not be required explicitly.

A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if its downward closure is an ideal. While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter.

Directed subsets are used in domain theory, which studies directed complete partial orders. These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of convergent sequences.