Disjointifiable lattice-ordered groups

This article studies disjointifiable lattice-ordered groups (abbr. dℓ-groups): the lattice-ordered groups G for which the frame \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}(G)$$\end{document} of all convex ℓ-subgroups is a normal frame; that is, for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A {\vee} B = G {\rm in} \mathcal{C}(G)$$\end{document} implies the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C, D {\in} \,{\mathcal{C}}(G)$$\end{document} such that C ⋂D = 0 and A ∨ D = C ∨ B = G. It is shown that if a Hahn group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V (\wedge, \mathbb{R})$$\end{document}) is a dℓ-group, then it is strongly disjointifiable (abbr. sdℓ), in the sense that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \vee B = G {\rm in} \mathcal{C}(G)$$\end{document} implies that there is a cardinal summand P of G, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P {\subseteq} A {\rm and}\,P^{\perp} {\subseteq} B$$\end{document}. Every finite valued ℓ-group is an sdℓ-group.