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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 $\mathcal P\to\mathcal P$)

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    Yes, sorry: commutativity and associativity.2012-09-16

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Some examples that are quite different from $a+b$ on $\mathbb N$ are $\min(a,b)$ and $\max(a,b)$, using a partial order on $M$ such that $a \le b$ implies $am \le bm$.

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    Can you please take a look at this question? http://math.stackexchange.com/questions/1693969/how-is-this-possible-to-convert-a-long-string-to-a-number-with-less-characters2016-03-12