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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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What else should your "+" satisfy? Commutativity, associativity, ...? –  Robert Israel Sep 16 '12 at 19:39
    
Yes, sorry: commutativity and associativity. –  Berci Sep 16 '12 at 20:54
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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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Thanks, good examples. Ok, let's say, I wanted "invertible" operation. Also, I am curious about all of these.. :) Moreover, I am also curious about the $\mathbb N$-type orders on $M$ which are compatible with the monoid structure. –  Berci Sep 16 '12 at 23:59
    
Note that one such partial order is known under the name "divides", and the corresponding min and max are known under the names gcd and lcm. –  celtschk Sep 17 '12 at 21:57
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