# A boolean representation of the Möbius function on a finite lattice

Let $(L,\wedge , \vee)$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$.

Consider the Möbius function $\mu$ on $L$ defined inductively by $$\mu(\hat{1}) = 1 \text{ and } \mu(a) = - \sum_{b>a}\mu(b)$$ Let $a_1 , \dots , a_n$ be the coatoms of $L$ and let $B_n$ be the rank $n$ boolean lattice, i.e. the subset lattice of $\{1,2, \dots , n\}$. Consider the map $m: B_n \to L$ defined by: $$m(I) = \bigwedge_{i \in I}a_i$$ Question: Is the following equality true? $$\mu(a) = \sum_{I \in m^{-1}(a)} (-1)^{|I|}$$

• This is essentially the Crosscut Theorem. See for example Enumerative Combinatorics, vol. 1, second ed., Corollary 3.9.4. – Richard Stanley Mar 8 '17 at 22:46
• Very nice! We just need to apply the Crosscut Theorem to the lattice $[a,\hat{1}]$ with $X$ the set of its coatoms. – Sebastien Palcoux Mar 8 '17 at 23:35