In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ,\leq, (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist c in A with a \leq c and b \leq c. A directed set's preorder is called a direction.
The notion defined above is sometimes called an . A is defined analogously, meaning that every pair of elements is bounded below.
Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward.
Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast ordered sets, which need