Face lattice

Example

Let $P \subset \R^d$ be a convex polytope, that is, the convex hull of finitely many points. A face of $P$ is $P$ itself, the empty set, or a set $P \cap H$ for a supporting hyperplane $H$. Here $H$ is supporting when $P$ lies in one closed half-space bounded by $H$. The face lattice of $P$ is the set of all faces of $P$, ordered by inclusion. It is a lattice: the meet of two faces is their intersection, and the join is the smallest face that contains both. It is graded by $\operatorname{rank}(F) = \dim F + 1$, with the empty face of rank $0$ and $P$ of rank $\dim P + 1$. It is an Eulerian poset.

Check for a square $Q$. The faces are the empty face, $4$ vertices, $4$ edges and $Q$ itself, so the lattice has $10$ elements. Two vertices have the empty face as meet; two adjacent vertices have their common edge as join, and two opposite vertices have $Q$ as join. In the interval from the empty face to $Q$, the elements of even rank are the empty face and the $4$ edges. The elements of odd rank are the $4$ vertices and $Q$. There are five of each, as the Eulerian condition asks.