Let be a free Abelian Semigroup, where is the unit element, and let be the Möbius Function. Then do the following properties,
See also Braun's Conjecture, Möbius Function
Flath, A. and Zulauf, A. ``Does the Möbius Function Determine Multiplicative Arithmetic?''
Amer. Math. Monthly 102, 354-256, 1995.