The Effros-Handelman-Shen-theorem tells you that Riesz groups are the same as dimension groups -- i.e. any ordered, unperforated abelian group with the Riesz interpolation property can be realised as the limit of a sequence $$ \ldots\rightarrow\mathbb{Z}^{d_{i-1}}\rightarrow^{\alpha_{i-1}} \mathbb{Z}^{d_i}\rightarrow^{\alpha_i}\mathbb{Z}^{d_{i+1}}\rightarrow^{\alpha_{i+1}}\ldots $$ of simplicial groups, i.e. where each $\mathbb{Z}^d$ has the standard ordering, and each $\alpha_i$ is a positive homomorphism.
Now, say you have a Riesz group (in my case, it could be $G=\mathbb{Z}[1/2]\oplus\mathbb{Z}$ with $G^+ = \{(a,b)\in G\, |\, a>0\}\cup\{0\}$). Is there an efficient way of writing down an explicit decomposition of this group as a limit of simplicial groups? I have not been able to find anything in the literature, which might mean that it is either very hard or completely trivial...