The hereditarily undecidability of one class of lattice-ordered Abelian groups

Let $G$ be an Abelian $\ell$-group. We denote through $G$ the lattice of all $\hat g$ with $g\in G$ (see [2]). Let $L$ be the class of all such Abelian $\ell$-group $G$ so that $G$ is divisible and Archimedean and $\hat G$ is atomic Boolean algebra.
\underline{THEOREM 1.} (Elementary) theory of $L$ is hereditarily undecidable.