Trying to follow the notes for understanding how to compute the zeta and mobius matrices from a graph. The graph is
Zeta matrix entries defined by
$\zeta(a,b)=\left\{\begin{matrix} 0 & if \quad a \nleqslant b \\ 1 & if \quad a \leq b \\ \end{matrix}\right.$
which gives zeta matrix $Z$ for rows and columns in order of $a,b,c,d$ (not sure how to get this, following containment $a \prec b \prec d$. This definition doesn't say anything about containment either.
$$Z=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$
For the Mobius matrix $M$, it is defined again without containment. The mobius function is defined
$$\mu(a,b)=\left\{\begin{matrix} 1 & if \quad a = b \\ 0 & if \quad a \nleqslant b \\ -\sum_{c:a \leq c \leq b}\mu(a,c) & if \quad a < b \\ \end{matrix}\right.$$
$$M=\begin{pmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$
It is clear that $ZM=I$, but i'm not sure how you get to containment/set relations from these numerical relations on the graph?