Incidence geometry

An incidence geometry is a structure which generalizes several concepts of incidence. Both abstract polytopes and hypertopes are 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$$

A chamber is a set of elements, $Y$, such that $$t(Y)=I$$ and any two elements in $Y$ are incident.

An incidence geometry is an incidence system such that for any set $F$, if all pairs of elements in $F$ are incident, then $F$ is a subset of some chamber.

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.