In

mathematics, a

**directed set** is a set equipped with a

transitive,

reflexive relation under which each pair of elements is bounded above. In other words, a set

**A** equipped with such a

relation--call it "≤"--is directed if whenever

**x** and

**y** are in

**A** there exists an element

**z** of

**A** which satisfies both

**x** ≤

**z** and

**y** ≤

**z**.

For example, the set of finite sets of integers, equipped with the partial order given by set inclusion, is directed: if **X** and **Y** are finite sets of integers, then their union **Z** is another finite set which contains each of them.