a top and bottom element. Therefore, a contractible intersection has Möbius function 0, an empty intersection has Möbius function $-1$ and $\mu(P(\eta)^o)=~\mu(\hat{\eta},\hat{\pi})$.

\[lem:inter\] If $S\subseteq \widehat{E}^{\sigma,\pi}$ and $|S|>1$ then: $$\mu(\bigcap_{\eta\in S}P(\eta)^o)=\begin{cases}-1,&\mbox{ if } S\in EZ^{\sigma,\pi}\\0,&\mbox{ otherwise }\end{cases}.$$

Let $\hat{\pi}=(\hat{\pi}_1,\ldots,\hat{\pi}_t)$ and define the *join* of $S$ to be $\vee S=(\max_{\eta\in S}(\hat{\eta}_1),\\\ldots,\max_{\eta\in S}(\hat{\eta}_t))$. The join is well defined because for each $i$ the set $\{\hat{\eta}_i\,|\,\eta\in S\}$ forms a chain, so there is an $\hat{\eta}_i$ that contains all others. The join of $S$ is the smallest element containing every embedding in $S$, so it is the bottom element of the intersection $I=\bigcap_{\eta\in S}P(\eta)^o$. Therefore, if $\vee S<\hat{\pi}$ then $I$ is contractible and so has Möbius function $0$, otherwise $\vee S=\hat{\pi}$ so $I$ is empty and thus has Möbius function $-1$. If $\vee S=\hat{\pi}$ this implies that every letter of $\pi$ is non-zero for some $\eta\in S$, that is, $S$ has empty intersection of zero sets, so $S\in EZ^{\sigma,\pi}$.

Consider our running example of $[132,413265]$. If $S^1=\{013200,\\010065\}$ then $013200$ decomposes to $(\emptyset,1,21,\emptyset)$ and $010065$ decomposes to $\\(\emptyset,1,\emptyset,21)$, so the join is: $$\vee S^1=(\max(\emptyset,\emptyset),\max(1,1),\max(21,\emptyset),\max(\emptyset,21))=(\emptyset,1,21,21).$$ Therefore, $\vee S^1<\hat{\pi}$ so the intersection is contractible. We can check this by looking at Figure \[fig:exA\] where we can see 