Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice.

Then $L$ is arithmetic
 * $\ll$ is multiplicative relation

where $\ll$ denotes the way below relation of $L$.

Sufficient Condition
Let $L$ be arithmetic.

Let $a, x, y \in S$ such that
 * $a \ll x$ and $a \ll y$

By Algebraic iff Continuous and For Every Way Below Exists Compact Between:
 * $\exists c \in K\left({L}\right): a \preceq c \preceq x$

and
 * $\exists k \in K\left({L}\right): a \preceq k \preceq y$

where $K\left({L}\right)$ denotes the compact subset of $L$.