If we have some function $f$ under $\mathbb{Z}$ and $$f(a, f(b, c)) = f(f(a, b), c)$$ $$f(a, b) = f(b, a)$$ $$f(a, 0) = a$$ $$f(a, -a) = 0$$ meaning $f$ is an abelian group with an identity element of $0$. Is that enough to prove that $f$ is addition? In other words, is an abelian group with a specific identity element unique within a certain domain? Similarly, could multiplication be proved if we had an abelian group with an identity element of $1$?
More generally, I'm trying to figure out if it is possible to deduce an algorithm (set of simple unambiguous recursive rules) from a set of axioms for some operation under a domain (axioms could perhaps be thought of as a form of ambiguous recursion). The first step is to figure out if a set of axioms can define the uniqueness of an operation. The more difficult step is to actually infer the specific recursive steps necessary.
No, this would imply that all (infinite) countable groups are isomorphic (as the underlying sets of any two such groups are in bijection), but that is not the case: For example both $(\mathbb{Z}, +)$ and $(\mathbb{Z} \times \mathbb{Z}, +)$ are countable, but they are not isomorphic, because the former is generated by a single element but the latter is not.