The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one

Let $G$ be an ordered group that is a direct sum of a rank-one torsion-free abelian group and a finite-rank torsion-free abelian group, with order structure arising from the natural order on the first summand. A necessary condition and a sufficient condition are given for $G$ to have an ordered-group inductive limit representation using injective maps.