On a certain problem of Ulam and its generalization

Let N be the set of all positive integers with usual operations a + b ≡ s(a, b) (addition) and a ∗ b ≡ m(a, b) (multiplication). For every bijection p : N × N → N define two mappings σ and μ from N to N such that σ(c) = σ(p(a, b)) = s(a, b) and μ(c) = μ(p(a, b)) = m(a, b) for all c ∈ N. Such a bijection p is known as Peano mapping. In [1, p. 32] S.M. Ulam asked “Does there exist a Peano mapping p such that addition commutes with multiplication in the sense that σ(μ(c)) = μ(σ(c)) for all c ∈ N ?”. No examples were known, and then the well-known Peano mapping p(a, b) = 2a−1(2b − 1) was seen by Ulam not to work, as μ(σ(2)) = μ(3) = 2, σ(μ(2)) = σ(2) = 3. Though the question is very natural and fascinating, it never got an answer since then.