Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-isomorphism
and $\Bbb N$ is a sub-semiring of $R$. Here, $\Bbb N$ is meant to include $0$. I do not think that there is such an extension, but I cannot find a contradiction. I also wonder if the problem becomes easiert when asking for an isomorphism
The multiplication will be commutative. Also, there will be many new "numbers", e.g. a unique additively absorbing element $\eta:=\varphi(0)$, i.e. $\eta+x=\eta$ for all $x\in R$. For $n\in\Bbb N^+$ we have
Therefore we have further elements $\tilde\eta=\varphi(\eta)$ that absorbe some numbers when added to them, e.g. $\varphi(n),n\in\Bbb N^+$, but not all (there can be only one universally absorbing element).