I am aware of three different constructions of the field of real numbers :
The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the completion of $\mathbb{Q}$ for the standard metric on $\mathbb{Q}$.
The Dedekind cut construction : in this case, the field $\mathbb{Q}$ is seen as a partially ordered set (with the standard order), and $\mathbb{R}$ is the completion of $\mathbb{Q}$ in the sense of the smallest complete lattice containing $\mathbb{Q}$.
The "Eudoxus" reals : in this case, we don't start from $\mathbb{Q}$ but directly from $\mathbb{Z}$. Reals numbers are identified as equivalent classes of "almost-homomorphisms" from $\mathbb{Z}$ into $\mathbb{Z}$ (functions $f$ from $\mathbb{Z}$ into $\mathbb{Z}$ such that $\{f(m+n)-f(m)-f(n): m, n\in\mathbb{Z}\}$ is finite and two almost-homomorphisms $f$, $g$ are equivalent iff the set $\{f(m)-g(m): m\in\mathbb{Z}\}$ is finite. As far as I understand, only the additive group structure on $\mathbb{Z}$ is used.
My question is this : We see that the first two constructions are actually just specific examples of application of a general process (metric completion, order completion). What is the "completion" process associated to the Eudoxus real construction (if there is one)? Could it be applied to other abelian groups, was is already explored ?