directed set

A directed set is a partially ordered set $(A,\le )$ such that whenever $a,b\in A$ there is an $x\in A$ such that $a\le x$ and $b\le x$.

A subset $B\subseteq A$ is said to be residual if there is $a\in A$ such that $b\in B$ whenever $a\le b$, and cofinal if for each $a\in A$ there is $b\in B$ such that $a\le b$.

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 $(A,\le )$ such that whenever $a,b\in A$ there is an $x\in A$ such that $x\le a$ and $x\le b$.

Note: Many authors do not require $\le$ 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.