  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.
