Let 
be a free Abelian semigroup, where 
is the identity element,
and let 
be the Möbius function. Define 
on the elements of the semigroup analogously to the
definition of 
(as 
if 
is the product of 
distinct primes) by regarding generators of the semigroup as primes. Then the Möbius
problem asks if the properties
1. 
implies 
for 
, where 
has the linear order 
,
2. 
for all 
,
imply that
 |
for all 
.
Informally, the problem asks "Is the multiplication law on the positive integers
uniquely determined by the values of the Möbius function and the property that
multiplication respects order?
The problem is known to be true for all 
if 
for all 
(Flath and Zulauf 1995).
