The interaction between addition and multiplication in the natural numbers has long been an interest of mine, leading to broader questions about the relationship between algebra and combinatorics. My gropings with those enigmas led me to the structures of Riffs and Rotes, extracting what we might think of as the “purely combinatorial” properties of primes factorizations. Thinking of the additive structure of the positive integers as embodied in their total linear ordering, the following two questions arise.
- How much of the natural ordering of the natural numbers is purely combinatorial?
- What additional axioms on the partial orders of Riffs and Rotes would restore their natural order?