From what I can gather the Free Commutative Monoid on infinite elements is isomorphic to the Natural Numbers under multiplication. The generating set is the prime numbers and every element of $\mathbb{N}^\times$ is uniquely defined by its prime factorisation $\prod p_i^{k_i}$.
I understand how a Free Commutative Monoid admits a partial ordering. First powers of a prime are ordered e.g. $2^0 < 2^1 < 2^2 < etc...$. Then the primes form a finite countable set which can be iterated so $2 < 3 < 5 < 7$ etc. Then the product of any primes is less than those individual primes: $ p_1 p_2 > p_2 > p_1$ (assuming ordered primes).
But how do you get from the Free Commutative Monoid on infinite elements to the total ordering of the natural numbers? e.g. why is $7 < (8 = 2^3)$ in $\mathbb{N}^\times$?