It is pointed out that it is possible to define the prime ideal spectrum of any ordered Abelian group with distributive ideal lattice, and that this provides a point of contrast between the Riesz interpolation property, which (at least in the case of a countable group) does not impinge on the spectrum, and the lattice property, which -- in the presence of the interpolation property, and the well-known local property that prime quotients are totally ordered -- is determined by the spectrum. (The determining property is that the intersection of two compact open subsets is compact).