# 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

### From first principles

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

1. There is a minimal element.
2. There is a maximal element.
3. Every totally ordered subset of π, called a chain, is contained within a totally ordered subset of π with exactly n+2  elements.
4. π is strongly connected.
5. 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, ${\displaystyle f, in π whose rank differs by 2, there are at least 2 elements h  such that ${\displaystyle f.

This last property is a weaker form of dyadicity.

## Realization

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. ${\displaystyle \mathbb {C} ^{n}}$ 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 ${\displaystyle \xi (x)=\varnothing }$.
• If x  is the maximal element of π then ${\displaystyle \xi (x)=X}$.
• For ${\displaystyle x,y\in {\mathcal {P}}}$, if ${\displaystyle x\lneq y}$ in π then ${\displaystyle \xi (x)\subsetneq \xi (y)}$.

## Related concepts

### Abstract polytopes

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

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