Why does the Möbius function take its values so often in $\{0,+1,-1\}$? The Möbius function of a locally finite poset $P$ is defined on its intervals $[x,y] \subseteq P$ recursively by
$$\mu([x,x])=1$$
$$\forall x < y : \mu([x,y])=-\sum_{x \leq z < y} \mu([x,z])$$
In many examples (see Wikipedia) it turns out that $\mu$ takes its values in $\{0,+1,-1\}$. Is there any conceptual reason for this? Also, is there some characterization of those posets $P$ for which this holds?
 A: Every one of the ones in that list which only take values $1,-1,0$ has the property that every range $[a,b]$ is isomorphic to some range in $\mathbb N^k$, the product of $k$ copies of the poset $\mathbb N$. 
We can show that the Möbius function for $P_1\times P_2$ is the product:
$$\mu_{P_1\times P_2}([(a_1,a_2),(b_1,b_2)]) = \mu_{P_1}([a_1,b_1])\mu_{P_2}([a_2,b_2])$$
So if you know that the Möbius function on $\mathbb N$ only takes values $0,1,-1$, you get that the Möbius functions for finite sets, finite multi-sets, natural numbers under divisibility, can only take values $0,1,-1$.
One key realization is that $\mu([a,b])$ is entirely determined by the isomorphism class of the sub-poset $[a,b]=\{x\in P\mid a\leq x\leq b\}$. Two intervals which are isomorphic as posets will be have the same value for the Möbius function, even if the intervals are in different posets.

Another view is that all of these posets are isomorphic to filters of the Multisets poset. So the values taken by each can only be the values take by Multisets.
