Buekenhout geometry

In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by .
Definition.
A Buekenhout geometry consists of a set "X" whose elements are called "varieties", with a symmetric reflexive relation on "X" called "incidence", together with a function τ called the "type map" from "X" to a set Δ whose elements are called "types" and whose size is called the "rank". Two distinct varieties of the same type cannot be incident.
A flag is a subset of "X" such that any two elements of the flag are incident. 
The Buekenhout geometry has to satisfy the following axiom:
Example: "X" is the linear subspaces of a projective space with two subspaces incident if one is contained in the other, Δ is the set of possible dimensions of linear subspaces, and the type map takes a linear subspace to its dimension. A flag in this case is a chain of subspaces, and each flag is contained in a so-called complete flag.
If "F" is a flag, the residue of "F" consists of all elements of "X" that are not in "F" but are incident with all elements of "F". The residue of a flag forms a Buekenhout geometry in the obvious way, whose type are the types of "X" that are not types of "F". A geometry is said to have some property residually if every residue of rank at least 2 has the property. In particular a geometry is called residually connected if every residue of rank at least 2 is connected (for the incidence relation).
Diagrams.
The diagram of a Buekenhout geometry has a point for each type, and two points "x", "y" are connected with a line labeled to indicate what sort of geometry the rank 2 residues of type {"x","y"} have as follows.