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? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

*Remark*: It is satisfied by any boolean lattice, more generally by the face lattice of any convex polytope (as suggested by John Shareshian), and more generally by any [Eulerian lattice][1].    
*Proof*: An Eulerian lattice is a graded lattice $L$ such that for any $a,b \in L$ with $a \le b$, we have $\mu(a,b)= (-1)^{|b|-|a|} $, with $a \mapsto |a|$ the rank function. The result is immediate. $\square$   

*Question 2*: Is there a non-Eulerian lattice with the above property on the Möbius function?  
**Yes**, see the answer of John Machacek.  

As suggested by Sam Hopkins:    
*Question 3*: Is there a non-Eulerian *atomistic* lattice with the above property on the Möbius function?  
*No* for $|L| \le 9$, as checked by the following Sage program:

    sage: n=0
    sage: for P in Posets():
    ....:     N=P.cardinality()
    ....:     if N>n:
    ....:         n=N
    ....:         print(n)
    ....:     if P.is_lattice():
    ....:         b = P.bottom()
    ....:         t = P.top()
    ....:         if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
    ....:             L=LatticePoset(P)
    ....:             if L.is_atomic():
    ....:                 if not L.is_graded():
    ....:                     print(P.cover_relations())
    ....:                 if L.is_graded():
    ....:                     for x in P:
    ....:                         for y in P:
    ....:                             if P.compare_elements(x,y)==-1:
    ....:                                 if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
    ....:                                     print(P.cover_relations())


*Remark*: This program is slow because it computes on the finite posets instead of the finite lattices. Any improvement are welcome.

 


  [1]: https://en.wikipedia.org/wiki/Eulerian_poset