The notion of [Eulerian lattice](https://en.wikipedia.org/wiki/Eulerian_poset) generalizes the notion of [face lattice](https://en.wikipedia.org/wiki/Convex_polytope#The_face_lattice) of a [convex polytope](https://en.wikipedia.org/wiki/Convex_polytope). 


(Bruggesse-Mani): The boundary complex of a convex polytope is [shellable](https://en.wikipedia.org/wiki/Shelling_(topology)).


(Björner-Wachs): A poset is said to be shellable if its [order complex](https://en.wikipedia.org/wiki/Poset_topology) is shellable. A bounded poset is shellable if and only if its proper part is shellable. 


([Wachs](https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwioyouOsPHZAhUpKsAKHZI-CeYQFggrMAA&url=http%3A%2F%2Fwww.math.miami.edu%2F~wachs%2Fpapers%2Ftoolnotes.pdf&usg=AOvVaw3k7rgAs9PEcIALvlrCeJkM)): The order complex of the face poset of a [simplicial complex](https://en.wikipedia.org/wiki/Abstract_simplicial_complex) is its [barycentric subdivision](https://en.wikipedia.org/wiki/Barycentric_subdivision).


*Pre-question*: Is the shellability kept by taking the barycentric subdivision?  

If so, the face lattice of a convex polytope is shellable. This leads to:


**Question**: Is an Eulerian lattice shellable?



Is an Eulerian lattice shellable?
The simplest example of a nonshellable (even non-Cohen Macaulay) Eulerian lattice is the disjoint union of two boolean algebras of rank three with their bottom elements identified and their top elements identified. The total number of elements is 14.


