%0 Journal Article %1 Martinez:2008 %A Mart\'ınez, Jorge %D 2008 %J Algebra Universalis %K L-Gruppen %N 1 %P 159--178 %R 10.1007/s00012-008-2092-y %T Disjointifiable lattice-ordered groups %U http://dx.doi.org/10.1007/s00012-008-2092-y %V 59 %X Abstract.~~This article studies disjointifiable lattice-ordered groups (abbr. dℓ-groups): the lattice-ordered groups G for which the frame \$\$\$\backslash$mathcal\C\\(G)\$\$ of all convex ℓ-subgroups is a normal frame; that is, for which \$\$A \$\backslash$vee\ B = G \$\backslash$rm in\ $\backslash$mathcal\C\(G)\$\$ implies the existence of \$\$C, D \$\backslash$in\ $\backslash$,\$\backslash$mathcal\C\\(G)\$\$ such that C ⋂D =~0 and A ∨D =~C ∨B =~G. It is shown that if a Hahn group \$\$V ($\backslash$wedge, $\backslash$mathbb\R\)\$\$) is a dℓ-group, then it is strongly disjointifiable (abbr. sdℓ), in the sense that \$\$A $\backslash$vee B = G \$\backslash$rm in\ $\backslash$mathcal\C\(G)\$\$ implies that there is a cardinal summand P of G, such that \$\$P \$\backslash$subseteq\ A \$\backslash$rm and\$\backslash$,P\^\$\backslash$perp\ \$\backslash$subseteq\ B\$\$. Every finite valued ℓ-group is an sdℓ-group. As should be expected, since these concepts are intrinsically frame-theoretic, their study at the level of frames should be fruitful. Indeed, for a frame embedding h: A → B whose adjoint satisfies the codensity condition that a ∨ b = 1 (in B) implies that $$h_*(a) h_*(b) = 1$$ (in A), we have that A is normal if and only if B is. Suitably interpreted for majorizing ℓ-subgroups H of G, this yields that H is a dℓ-group (resp. sdℓ-group) precisely when G has the property.

@article{Martinez:2008, abstract = {Abstract.~~This article studies disjointifiable lattice-ordered groups (abbr. dℓ-groups): the lattice-ordered groups G for which the frame {\$}{\$}{\{}{$\backslash$}mathcal{\{}C{\}}{\}}(G){\$}{\$} of all convex ℓ-subgroups is a normal frame; that is, for which {\$}{\$}A {\{}{$\backslash$}vee{\}} B = G {\{}{$\backslash$}rm in{\}} {$\backslash$}mathcal{\{}C{\}}(G){\$}{\$} implies the existence of {\$}{\$}C, D {\{}{$\backslash$}in{\}} {$\backslash$},{\{}{$\backslash$}mathcal{\{}C{\}}{\}}(G){\$}{\$} such that C ⋂D =~0 and A ∨D =~C ∨B =~G. It is shown that if a Hahn group {\$}{\$}V ({$\backslash$}wedge, {$\backslash$}mathbb{\{}R{\}}){\$}{\$}) is a dℓ-group, then it is strongly disjointifiable (abbr. sdℓ), in the sense that {\$}{\$}A {$\backslash$}vee B = G {\{}{$\backslash$}rm in{\}} {$\backslash$}mathcal{\{}C{\}}(G){\$}{\$} implies that there is a cardinal summand P of G, such that {\$}{\$}P {\{}{$\backslash$}subseteq{\}} A {\{}{$\backslash$}rm and{\}}{$\backslash$},P\^{}{\{}{$\backslash$}perp{\}} {\{}{$\backslash$}subseteq{\}} B{\$}{\$}. Every finite valued ℓ-group is an sdℓ-group. As should be expected, since these concepts are intrinsically frame-theoretic, their study at the level of frames should be fruitful. Indeed, for a frame embedding h: A → B whose adjoint satisfies the codensity condition that a ∨ b = 1 (in B) implies that $$h_{*}(a) \vee h_{*}(b) = 1$$ (in A), we have that A is normal if and only if B is. Suitably interpreted for majorizing ℓ-subgroups H of G, this yields that H is a dℓ-group (resp. sdℓ-group) precisely when G has the property. }, added-at = {2013-02-02T14:43:22.000+0100}, author = {Mart{\'\i}nez, Jorge}, bdsk-file-1 = {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}, bdsk-url-1 = {http://dx.doi.org/10.1007/s00012-008-2092-y}, biburl = {https://www.bibsonomy.org/bibtex/256165fc2d520b092a1d5e0f1aab78fbe/ks-plugin-devel}, date-added = {2009-10-26 14:24:04 +0100}, date-modified = {2009-10-26 14:24:04 +0100}, day = 01, doi = {10.1007/s00012-008-2092-y}, groups = {public}, interhash = {9025aa13af9ab95c80e1bab3c5ca305c}, intrahash = {56165fc2d520b092a1d5e0f1aab78fbe}, journal = {Algebra Universalis}, keywords = {L-Gruppen}, month = {11}, number = 1, pages = {159--178}, timestamp = {2013-02-02T14:43:22.000+0100}, title = {Disjointifiable lattice-ordered groups}, ty = {JOUR}, url = {http://dx.doi.org/10.1007/s00012-008-2092-y}, username = {keinstein}, volume = 59, year = 2008 }