Disjointifiable lattice-ordered groups

This article studies disjointifiable lattice-ordered groups (abbr. dl-groups): the lattice-ordered groups G for which the frame C(G) of all convex l-subgroups is a normal frame; that is, for which A boolean OR B = G in C(G) implies the existence of C, D is an element of C(G) such that C boolean AND D = 0 and A boolean OR D = C boolean OR B = G. It is shown that if a Hahn group V (Lambda, R) is a dl-group, then it is strongly disjointifiable (abbr. sdl), in the sense that A boolean OR B = G in C(G) implies that there is a cardinal summand P of G, such that P subset of A and P-perpendicular to subset of 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 boolean OR b = 1 (in B) implies that h(*) (a) boolean OR 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.