\begin{definition}[Definition:Arithmetic Ordered Set]
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.
$L$ is '''arithmetic''' {{iff}}
:$L$ is algebraic and $K\left({L}\right)$ is meet closed
where $K\left({L}\right)$ denotes the compact subset of $L$.
\end{definition}