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.