A partial order is directed complete (or a dcpo) just if it has least upper bounds for every directed subset.
Let be a nonempty subset of . is directed whenever every pair has an upper bound in , i.e. an element such that and .
Directedness means that every pair of elements of carries compatible information: there is at least one element that is more informative than both (and hence, the information that and carry is not contradictory).
Thus, is "going somewhere": its elements can be seen as approximants that can be combined into better approximants. Directed-completeness means that every such "set with a purpose" has a least upper bound, i.e. a least informative element fitting the "specification" given by all the approximants.