R×R上到R的某类1-1映射的不存在性

In p.32 of 1] the following problem is posed: A problem on Peano mapping Let R be the set of positive rational integers with usual operation a+b≡s(a,b) and a·b≡m(a,b). Every one-to-one(Peano) mapping c=p(a,b) on R×R to all R may serve so associate with s(a,b) and m(a,b) two functions σand μ on R to R by the definitions σ(c)=σ(p(a,b))= s(a,b), and μ(c)=μ(p(a,b))= m(a,b). Does there exist a Peano mapping p(a,b) such that "addition commutes with multiplication" in the sense that σ(μ(c))=μ(σ(c))for all c of R? To illustrate, we note