In mathematics, a Buekenhout geometry or diagram geometry is a generalization of

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

Tits building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Bui ...

s, 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 A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...

is a subset of ''X'' such that any two elements of the flag are incident. The Buekenhout geometry has to satisfy the following axiom: *Every flag is contained in a flag with exactly one variety of each type. Example: ''X'' is the

linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...

s of a

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

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 have as follows. *If the rank 2 residue is a digon, meaning any variety of type ''x'' is incident with every variety of type ''y'', then the line from ''x'' to ''y'' is omitted. (This is the most common case.) *If the rank 2 residue is a projective plane, then the line from ''x'' to ''y'' is not labelled. This is the next most common case. *If the rank 2 residue is a more complicated geometry, the line is labelled by some symbol, which tends to vary from author to author.

References

* * * * *

External links

*{{Commonscat-inline Incidence geometry Group theory Algebraic combinatorics Geometric group theory