A problem of Ulam

Let $R$ be a set of positive integers with usual operations of addition and multiplication
$$ a+b=s(a,b);\quad a\cdot b=m(a,b);\quad a,b\in R. $$
A correspondence is set up between each one-to-one (Peano) mapping $p$ of the space $R\times R$ onto the whole of $R$ and the two functions
$$ \begin{aligned} \sigma(c)&=\sigma[p(a,b)]=s(a,b);\\ \mu(c)&=\mu[p(a,b)]=m(a,b). \end{aligned} $$
It is proved in this note that there can be no Peano mapping for which $\sigma(\mu(c))=\mu(\sigma(c))$ for all $c$ in $R$.