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.

Incidence system
An incidence system is a tuple $$(X,*,t,I)$$ where In the context of incidence system a flag is a set of elements, $X$, such that any two elements in $I$ are incident. This is a distinct notion from the flag of a polytope.
 * $$ is a set whose members are called elements.
 * $X$ is a finite set whose members are called types.
 * $$t : X\rightarrow I$$ is a type function, mapping each element to a type.
 * $F$ is a binary relation on $F$ called incidence. It obeys 3 properties
 * $$\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$$

A chamber is a flag, $F$, such that $$t(F)=I$$.

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 $$I$$.

Incidence structure
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 $$\left(P,L,I\right)$$ 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.

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.