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:

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

Directed complete partial orders

Partially ordered sets