Given $\frak M$ a countable nonstandard model of $\sf PA$ and let $a\in M$ be a nonstandard element. A "cluster around $a$" is the set of successors and predecessors of $a$, a cluster is a subset of $M$ that is a cluster around some $a\in M$.

It is standard exercise to show that the set of clusters of $\frak M$ is order-isomorphic to $\mathbb Q$, and it is simple to see that it is not isomorphic to $\Bbb Q^+$ when adding the additive structure (this can be seen by looking at the bounded sequence $(nX\mid n\in\Bbb N)$ for some cluster $X$).

Indeed the additive structure of the clusters of $\frak M$ seems to be quite complicated, the similarity to $\Bbb Q^+$ doesn't even hold "locally" (in the sense that given any cluster $X$ there exists a cluster $Y$ such that $Y$ is bigger than $X$ and smaller than any $X+X/n$ for standard $n$ $^1$).

So the ordered additive structure of the clusters of $\frak M$ is quite complicated, which brings me to the following questions:

1. Given $(C,<,+)$ the structure of clusters of $\frak M$, can we recover the additive structure on $M$?

2. If the answer of the above is "yes", what about uncountable models?

$^1$ Note that division by a standard number is defined as $X/n$ is the unique cluster $Y$ such that there is $y∈Y$ and $x∈X$ with $yn=x$