Semigroup isomorphism between $(\{1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$. I know that the two semigroups $(\{0,1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$ are not isomorphic because if we want to map identity elements together then it can be see that  we can't have injective function between them,but what can we say about $(\{1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$?

Suppose there is an isomorphism $f:(\Bbb{N},+) \to ((\Bbb{N}-\{0\}, \times)$. Then since $f$ preserves idempotents, one has $f(0) = 1$. Let $a = f(1)$. Then for every $n >0$, $f(n) = a^n$. Thus $f(\Bbb{N}) = \{a^n \mid n \geqslant 0\}$ and hence $f$ is not a bijection, a contradiction.
