## Möbius Problem

Let be a free Abelian Semigroup, where is the unit element, and let be the Möbius Function. Then do the following properties,

1. Implies for , where has the linear order ,

2. for all ,
imply that

for all ? The problem is known to be true for for all .

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