# Incidence geometry

An **incidence geometry** or **Buekenhout geometry** is a structure which generalizes several concepts of incidence. Abstract polytopes, configurations and hypertopes are all incidence geometries.

## Definition[edit | edit source]

### Incidence system[edit | edit source]

An **incidence system** is a tuple where

- X is a set whose members are called
*elements*. - I is a finite set whose members are called
*types*. - is a
*type function*, mapping each element to a type. - * is a binary relation on X called
*incidence*. It obeys 3 properties- (reflexivity)
- (symmetry)

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.

A **chamber** is a flag, F, such that .

### Incidence geometry[edit | edit source]

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 .

## Incidence structure[edit | edit source]

A incidence system with two types is called an **incidence structure**. Incidence structures are generally not rich enough to make them interesting objects of study on their own. However they are used as a starting point for more rich objects such as configurations, generalized polygons and near polygons.

An incidence structure may be defined more simply as a triple 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[edit | edit source]

### Abstract polytopes[edit | edit source]

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.

## External links[edit | edit source]

- Wikipedia Contributors. "Buekenhout geometry".
- Wikipedia Contributors. "Incidence structure".

## Bibliography[edit | edit source]

- Fernandes, Maria (2014). "Regular and chiral hypertopes" (PDF).
- Fernandes, Maria; Leemans, Dmitri; Weiss, Asia (2016). "Highly symmetric hypertopes".
*Aequationes mathematicae*.