On a certain problem of Ulam and its generalization (Q988499)

Let \(\mathbb N\) be the set of positive integers. A Peano mapping is a bijection \(p:{\mathbb N}\times {\mathbb N}\to {\mathbb N}\) for which there are defined two mappings \(\sigma, \mu:{\mathbb N} \to {\mathbb N}\) by \(\sigma=s\circ p^{-1}\) and \(\mu=m\circ p^{-1}\), where \(s,m:{\mathbb N}\times {\mathbb N}\to {\mathbb N}\), \(s(a,b)=a+b\), \(p(a,b)=a\cdot b\). The author proves that there exists no Peano mapping \(p\) such that \(\sigma\circ\mu= \mu\circ\sigma\).