A Noncommutative Version of the Natural Numbers

In this note, we construct and study an algebraic system similar to the natural numbers, but with noncommutative addition. The addition we introduce is a binary operation that commutes with itself in the sense of N. Durov. Neverheless, the multiplication in this system (defined by iterating the noncommutative addition) turns out to be associative, commutative, and distributive over addition, and the resulting system has interesting and nontrivial arithmetic.