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
References
Flath, A. and Zulauf, A. ``Does the Möbius Function Determine Multiplicative Arithmetic?''
Amer. Math. Monthly 102, 354-256, 1995.