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Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure

There is an extension $R$: take the closure of $\mathbb N$ by the operations $\text{L}$ (or $\varphi$ in the OP) and its inverse $\text{E}$, which are the logarithm and exponential in base $1.2$. …
Yaakov Baruch's user avatar