Conrad frames

Abstract

A Conrad frame is a frame which is isomorphic to the frame C ( G ) of all convex ℓ-subgroups of some lattice-ordered group G. It has long been known that Conrad frames have the disjointification property. In this paper a number of properties are considered that strengthen the disjointification property; they are referred to as the Conrad conditions. A particularly strong form of the disjointification property, the C-frame condition, is studied in detail. The class of lattice-ordered groups G for which C ( G ) is a C-frame is shown to coincide with the class of pairwise splitting ℓ-groups. The arguments are mostly frame-theoretic and Choice-free, until one tackles the question of whether C-frames are Conrad frames. They are, but the proof is decidedly not point-free. This proof actually does more: it shows that every algebraic frame with the FIP and disjointification can be coherently embedded in a C-frame. When the discussion is restricted to normal-valued lattice-ordered groups, one is able to produce examples of coherent frames having disjointification, which are not Conrad frames.