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Consider a $n$ by $n$ by $n$ grid represented by the set of $3$-uples $S=\{1,2,\dots, n\}^3$. A line (resp. slice) of $S$ is a subset of cardinal $n$ (resp. $n^2$) where two components (resp. one ...

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$. Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...