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Existence of a non-Eulerian atomistic lattice with this property on the Möbius function

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},...
Sebastien Palcoux's user avatar