In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by Buekenhout (1979).
Two distinct varieties of the same type cannot be 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.
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.
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.