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Prove that the monoids $(\mathbb N,+)$ and $(\mathbb N,\cdot)$ aren't isomorphic.

I tried that by assuming that there is an isomorphism f between (N,+) and (N,*). Then f(x+y)=f(x)*f(y), for every x,y from N and f(0)=1. For x=0 and y=1 we have f(1)=f(0)*f(1). But f(0)=1. That means that f(1)=f(1), so it doesn't help me with anything.

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    $\begingroup$ morphisms as what algebraic structure - semigroups? $\endgroup$
    – Petar Ivanov
    Nov 22, 2015 at 10:35
  • $\begingroup$ @PetarIvanov Judging from what he has written, I'm quite sure he means monoids. $\endgroup$
    – Stefan Perko
    Nov 22, 2015 at 11:07

1 Answer 1

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Let $f$ is the isomorphism.
Let $f(1)= c$.
Then using induction $f(n) = f(1)^n = c^n$, which can't be bijection (since it's not onto).

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  • $\begingroup$ Thank you very much for your help! $\endgroup$
    – Adrian Berteanu
    Nov 22, 2015 at 10:42

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