# Incidence complex

An **incidence complex** is a combinatorial structure that generalizes the incidence of polytopes. It is a refinement of the earlier polystroma and the precursor to the later abstract polytope.

## Definition[edit | edit source]

### From first principles[edit | edit source]

An incidence complex π of rank n , also called an **n -complex**, is a partially ordered set with the following properties:

- There is a minimal element.
- There is a maximal element.
- Every totally ordered subset of π, called a
**chain**, is contained within a totally ordered subset of π with exactly n+2 elements. - π is strongly connected.
- Let the rank of an element e be two less than the size of a maximal chain such that e is its maximal element. For a pair of elements, , in π whose rank differs by 2, there are at least 2 elements h such that .

This last property is a weaker form of dyadicity.

## Realization[edit | edit source]

This section needs expansion. You can help by adding to it. |

Both complex and quaternionic polytopes are realizations of incidence complexes in a unitary space following the same rules:

Let π be an incidence complex. Let X be a unitary space (e.g. for complex polytopes). Then let ΞΎ be a mapping from elements of π to affine subspaces of X .

- If x is the minimal element of π then .
- If x is the maximal element of π then .
- For , if in π then .

## Related concepts[edit | edit source]

### Abstract polytopes[edit | edit source]

The definition of an incidence complex is identical to that of abstract polytopes except for that it uses a weaker version of dyadicity. Abstract polytopes are a required to have *exactly* 2 elements between elements whose ranks differ by 2, while incidence complexes only require *at least* 2 elements.

## Bibliography[edit | edit source]

- Schulte, Egon (2018),
*Regular Incidence Complexes, Polytopes, and C-Groups*, arXiv:1711.02297v1