Understanding the values of Möbius function

Question

I want to understand how did he get the values for this intersection poset:

So I know that he used the Möbius function but I don't understand how he get the values only using the following :

But how did he end up with negative values? If someone can explain to me thanks in advance and why for $tuv$ he gets $2$ not $3$

Answer 1

If you transpose the $\mu(x,y)$ term, the condition that

$$\sum_{z\in[x,y]}\mu(x,z)=0$$

for $x<y$ becomes

$$\mu(x,y)=-\sum_{z\in[x,y)}\mu(x,z)\,.$$

In his example, for instance,

$$\mu(u)=\mu(\hat 0,u)=-\sum_{z\in[\hat 0,u)}\mu(\hat 0,z)=-\mu(\hat 0,\hat 0)=-1\,,$$

since $\hat 0$ is the only element of $[\hat 0,u)$.