Buekenhout geometry

Remarks

A flag $C$ with $\tau(C) = I$ is a chamber; by clause 1 it has exactly one element of each type. The number $\abs{I}$ is the rank. A geometry of rank $2$ is the same as an incidence structure whose flags extend, such as a projective plane or a generalized polygon. The residue of a flag $F$ is the geometry of the elements incident with all of $F$ and not in $F$, over the types not met by $F$. The Buekenhout diagram records, for each pair of types $i \ne j$, the class of the rank $2$ residues of type $\set{i, j}$. Tits buildings give geometries whose diagrams are Coxeter diagrams.

Example: let $\mathcal{P}$ be an $n$-dimensional projective space. Take $X$ to be the set of its subspaces of dimension $0, 1, \dots, n - 1$, let $\tau$ be the dimension, and let $x * y$ mean $x \subseteq y$ or $y \subseteq x$. Clause 1 holds since two distinct subspaces of the same dimension are not contained in each other. Clause 2 holds since every chain of subspaces can be refined to a chain with one subspace of each dimension.

Non-example: take the lines of a projective plane, all of one type, with $x * y$ meaning that $x$ and $y$ meet. Any two lines meet, so two distinct incident elements have the same type and clause 1 fails.