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
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.
 * $$t : X\rightarrow I$$ is a type function, mapping each element to a type.
 * $$ is a binary relation on $X$ called incidence. It obeys 3 properties
 * $$\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$$

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 $$t(F)=I$$.

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 $$I$$.

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.