Improper element

An element of a polytope is called improper whenever it's the minimal or maximal element of the polytope. All other elements are said to be proper.

Traditionally, these elements aren't considered. However, taking them into account is mathematically useful in a lot of circumstances. If we were to ignore them, the following annoyances would occur:
 * Stating the diamond condition would require us to special-case that edges have two vertices and ridges join two facets.
 * Convex polytopes would no longer form a lattice.
 * Polytope products would have a much messier formal definition.
 * The nullitope and the point would no longer be distinct.
 * The n-simplex would no longer have $$2^{n+1}$$ elements, instead having exactly two less.