Question: Consider the binary operation on the free commutative magma generated by one element. Is this binary operation also associative?
Note: I do not mean the free magma (1)(2) generated by one element -- I mean the quotient of this object by the appropriate equivalence relation inducing commutativity.
Background: This question is meant as a sort of "converse" to this question. In the answers and discussion surrounding that question, it was established that the associativity of addition of $\mathbb{N}$ implies the commutativity of the addition of $\mathbb{N}$ (see also here, here, or here). More generally, it was established that the binary operations of both the free semigroup and of the free monoid generated by one element are commutative. (Obviously these are isomorphic to $\mathbb{N}$, depending on whether one doesn't or does include $0$ as a natural number.)
In particular, I am well aware that, in general, neither does associativity imply commutativity nor does commutativity imply associativity. Neither general implication is the nature of my question.
As far as I understand, it is also possible to prove that addition on the natural numbers is commutative without appealing to associativity (1)(2). However, it can also be proven by appealing to associativity. So I would be interested to know if there is a proof of associativity of addition on the natural numbers which appeals to commutativity of addition (assuming that commutativity can be proven independently of associativity).
