Consider the (multiplically written) free commutative monoid $M$ on a countably infinite set $\mathcal P$ of generators (it is isomorphic to $(\mathbb N,\cdot)$ with the primes as generators, $\mathcal P:=\{2,3,5,7,11,13,\ldots\}$).
Q1: Are all those commutative, associative $+$ operations described on $M$ somewhere in the literature which satisfy the distributive law ($(a+b)m = am+bm$)? We can restrict first to the cancellative $+$ operations.
Q2: Is it true that each of these can be obtained by some automorphism $M\to M$ (i.e. using a permutation $\mathcal P\to\mathcal P$)