Improper element

The Hasse diagram (right) of a square pyramid. U and Ø are the improper elements.

An element of a polytope is called improper whenever it's the minimal or maximal element of the polytope. These elements can be thought of as the empty set and the entire polytope. All other elements are said to be proper.

Traditionally, these elements aren't considered. Nevertheless, taking them into account is mathematically convenient 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.
• The nullitope and the point would no longer be distinct.
• Polytope products would have much messier formal definitions.
• The n-simplex would no longer have $2^{n+1}$ elements, instead having exactly two less.

There are also circumstances where considering improper elements would be mathematically convenient but for historical reasons they are still not used:

• The Euler characteristic of a spherical polytope is 2 if the rank is odd and 0 if the rank is even. If improper elements were considered the Euler characteristic of a spherical polytope would always be 0.
• The genus of a polyhedron can be calculated as $1-\chi/2$ for orientable polyhedra and $2-\chi$ for non-orientable polyhedra. If improper elements were considered these would simplify to $-\chi/2$ and $-\chi$ respectively.

Despite all of this, one can generalize polytopes in a way that completely ignores the improper elements, as hypertopes.