Braun's Conjecture

Let $B=\{b_1, b_2, \ldots\}$ be an Infinite Abelian Semigroup with linear order $b_1<b_2<\ldots$ such that $b_1$ is the unit element and $a<b$ Implies $ac<bc$ for $a,b,c\in B$. Define a Möbius Function $\mu$ on $B$ by $\mu(b_1)=1$ and

$$\sum_{b_d\vert b_n} \mu(b_d)=0$$

for $n=2$, 3, .... Further suppose that $\mu(b_n)=\mu(n)$ (the true Möbius Function) for all $n\geq 1$. Then Braun's conjecture states that

$$b_{mn}=b_m b_n$$

for all $m,n\geq 1$.

See also Möbius Problem

References

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