Definable relation

Question

How can we show that the addition relation $\{\langle m,n,p\rangle\mid p=m+n\}$ is not definable in $(\mathbb{N};\space \cdot)$ (structure with universe $\mathbb{N}$ and the usual multiplication)?

Answer 1

Consider the automorphism $h:\mathbb{N} \rightarrow\mathbb{N} $ that is the identity map except for multiples of some distinct prime numbers a, b, where, in any such $n$, we replace each a by b, and each b by a (e.g., if $n=2*a*a*b$, then $ h(n)=2*b*b*a$).

For any m-ary relation R definable in $\mathbb{N}$, and any $a_{1}, ..., a_{m}\in\mathbb{N}$, we have $\langle a_{1}, ..., a_{m}\rangle\in R \iff \langle h(a_{1}), ..., h(a_{m})\rangle\in R$. By assuming that addition is definable, we arrive at a contradiction.