Let $G$ be an Archimedean $\ell $-group. We denote by $G^d$ and $R_D(G)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation $(R_D(G))^d=R_D(G^d)$. As an application, we show that if $G$ is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.
Lattices in the class TRN of algebraic, distributive lattices whose compact elements form relatively normal lattices are investigated. We deal mainly with the lattices in TRN the greatest element of which is compact. The distributive radicals of algebraic lattices are introduced and for the lattices in TRN with the sublattice of compact elements satisfying the conditional join-infinite distributive law they are compared with two other kinds of radicals. Connections between complete distributivity of algebraic lattices and the distributive radicals are described. The general results can be applied e.g. to MV -algebras, GMV -algebras and unital l-groups.