Abbreviation: DCPO

A \emph{directed complete partial order} is a poset $\mathbf{P}=\langle P,\leq \rangle$ such that every directed subset of $P$ has a least upper bound: $\forall D\subseteq P\ (D\ne\emptyset\mbox{and}\forall x,y\in D\ \exists z\in D (x,y\le z)\Longrightarrow \exists z\in P(z=\bigvee D))$.

Let $\mathbf{P}$ and $\mathbf{Q}$ be directed complete partial orders. A morphism from $\mathbf{P}$ to $\mathbf{Q}$ is a function $f:Parrow Q$ that is \emph{Scott-continuous}, which means that $f$ preserves all directed joins:

$z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$

Example 1: $\langle \mathbb{R},\leq \rangle$, the real numbers with the standard order. Example 1: $\langle P(S),\subseteq \rangle$, the collection of subsets of a sets $S$, ordered by inclusion.

$\begin{array}{lr} f(1)= &1 f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & \end{array}$

Complete semilattices

Directed partial orders