Abbreviation: DPO
A \emph{directed partial order} is a poset $\mathbf{P}=\langle P,\leq \rangle $ that is \emph{directed}, i.e. every finite subset of $P$ has an upper bound in $P$, or equivalently, $P\ne\emptyset$, $\forall xy\exists z (x\le z$ and $y\le z)$.
Let $\mathbf{P}$ and $\mathbf{Q}$ be directed partial orders. A morphism from $\mathbf{P}$ to $\mathbf{Q}$ is a function $f:Parrow Q$ that is order preserving:
$x\le y\Longrightarrow f(x)\le f(y)$
Example 1:
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &2
f(4)= &
f(5)= &
f(6)= &
\end{array}$