Abbreviation: APos

An \emph{algebraic poset} is a directed complete partial orders $\mathbf{P}=\langle P,\leq \rangle$ such that

the set of compact elements below any element is directed and

every element is the join of all compact elements below it.

An element $c\in P$ is \emph{compact} if for every subset $S\subseteq P$ such that $c\le\bigvee S$, there exists a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$.

The set of compact elements of $P$ is denoted by $K(P)$.

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

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

Example 1:

$\begin{array}{lr} f(1)= &1 \end{array}$

Algebraic semilattices

Directed complete partial orders