How many maps $\phi : \mathbb{N}\cup\{0\} \to \mathbb{N}\cup\{0\}$ are there, with the property that $\phi(ab) = \phi(a) + \phi(b)$, $\forall a, b$ $\in \mathbb{N} \cup \{0\}$?
Can I get a hint to solve this problem?
I have noticed that $\phi(0) = 0$ because $\phi(0\cdot0) =2\phi(0) \implies \phi(0) =0$, $\phi(a\cdot0)= \phi(a)+\phi(0)\implies \phi(0)=\phi(a)$
So one of the map is one which sends every element to 0. How to check if there are other maps satisfying the above condition?