Simple Riesz groups of rank one having wild intervals

We prove that every partially ordered simple group of rank one which is not Riesz embeds into a simple Riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals. Using this result, we construct examples of simple Riesz groups of rank one G, containing unbounded intervals (Dn)n⩾1 and D, that satisfy: (a) for each n⩾1, tDn≠G+ for every (t