# Incidence geometry

(Redirected from Incidence system)

An incidence geometry or Buekenhout geometry is a type of combinatorial structure comprising a typed set of elements with an incidence relation describing how they are connected. Incidence geometries are highly general and encompass abstract polytopes, configurations, hypertopes, generalized polygons, and many other concepts.

## Definition

### Incidence system

An incidence system is a tuple $(X,*,t,I)$ where

• X  is a set whose members are called elements.
• I  is a finite set whose members are called types. (Types generalize the notion of rank in polytopes, or the distinction between nodes and edges in a graph.)
• $t:X\rightarrow I$ is a type function, mapping each element to a type.
• *  is a binary relation on X  called incidence. If $x*y$ is true, we say that x  and y  are incident. The relation must obey three properties:
• $\forall x.x*x$ (reflexivity)
• $\forall x,y.x*y\iff y*x$ (symmetry)
• $\forall x,y.x*y\land t(x)=t(y)\implies x=y$ . This states that two distinct elements of the same type cannot be incident (for example, in a graph, two vertices are not considered incident).

In the context of incidence system a flag is a set of elements, F , such that any two elements in F  are incident. This is a distinct notion from the flag of a polytope. It follows from the final condition on the incidence relation that all the types of elements in a flag must be distinct.

A chamber is a flag, F , such that $t(F)=I$ (using the image). In a chamber, every type is represented exactly once.

### Incidence geometry

An incidence geometry is an incidence system such that every flag is a subset of some chamber. In other words, every maximal clique in the incidence graph has type $I$ . A necessary but not sufficient condition is that every element must be incident on elements of every other type.

## Incidence structure

A incidence system with two types is called an incidence structure. Incidence structures in general are too broad of a category to make them interesting objects of study on their own, but several subclasses of incidence structures have interesting behavior:

If every element in an incidence structure is incident on at least one other distinct element, then it is also an incidence geometry. For example, a graph is an incidence geometry iff every vertex has a degree of at least 1.

An incidence structure may be defined more simply as a triple $\left(P,L,I\right)$ where P  is a set of vertices, L  is a set of edges and I  is an incidence relation. Some authors require that I  is symmetric, however this is only required of you do not require I  to be typed.

## Examples

### Abstract polytopes

Every abstract polytope is an incidence geometry.

• The element set are the elements of the abstract polytope.
• The incidence relation is incidence.
• The type function is the rank of each element.
• The chambers are flags of the abstract polytope.