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Keywords:
Archimedean $\ell $-group; divisible hull; distributive radical; complete distributivity
Archimedean $\ell $-group; divisible hull; distributive radical; complete distributivity
Summary:
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.
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