MATH.1:5.2 - Natural numbers from repetition

Take one object X and one generating loop a:X→X. The paths are the empty word, a, a;a, and longer repetitions. Write a^n for the list containing n copies, with a^0=id_X.

Concatenating a^m and a^n gives a^(m+n). Thus lengths supply an arithmetic account of this structure: the identity corresponds to 0 and composition corresponds to addition. Every path is determined by its length in this one-generator case.

Adding a second loop b changes the situation. The paths a;b and b;a both have length 2 but are different lists. Counting generators now loses their order. It still answers a length question; a question about which generator acts first requires the path.