Topological abelian groups with a Dedekind lattice of closed subgroups

An investigation of the structure of locally compact abelian groups whose lattices of closed subgroups satisfy Dedekind's axiom. This condition is closely related to the finite-rank condition. In contrast to the discrete case, Dedekind groups form a relatively small subclass of the class of all locally compact abelian groups. A description is given of abelian groups such that the product of any two closed subgroups is closed.