Directed-complete partial order

Remarks

A dcpo with a least element $\bot$ is called pointed, and is often simply called a cpo. Example: the set of partial functions from $\N$ to $\N$, ordered by inclusion of graphs. If $D$ is a directed family, any two members agree where both are defined, so $\bigcup D$ is again a partial function and is the supremum. Non-example: $\N$ with its usual order is not a dcpo, because $\N$ itself is directed and has no upper bound. Every finite poset is a dcpo, since a finite directed set has a greatest element. Equivalently, by a theorem that uses the axiom of choice, a poset is a dcpo if and only if every nonempty chain has a supremum.