The structure of lattice-ordered commutative topological groups of compact origin

It is shown that every lattice-ordered commutative separable topological group of compact origin can be obtained from a finite number of its linearly ordered subgroups, each of which is isomorphic either to the additive group of real numbers with the natural topology and the usual order or to a subgroup of the additive group of real numbers with the discrete topology and the usual order, admitting a finite system of linearly independent generators, by forming in turn the direct and the lexicographic products.