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BuekenhoutTits geometry
A BuekenhoutTits geoemtry, or simply geometry, is a quadruple $(\Gamma,\#,T,\tau)$, where
1. 2. $T$ is a set whose elements are called types
3. 4. $\#$ is a reflexive symmetric relation on $\Gamma$ (called the incidence relation) such that if objects $a,b$ are of the same type and $a\#b$, then $a=b$.
We usually write $\Gamma$ for the geometry. The cardinality of $T$ is called the rank of $\Gamma$. $T$ is usually assumed to be finite.
Here’s a simple example of a BuekenhoutTits geometry. Let $T$ be the set consisting of four types: vertex, edge, face, and cube. Let $X=\{A,B,C,D,E,F,G,H\}$, and $\Gamma$ be the set of subsets of $X$ consisting of the following:

all of the singletons

the following doubletons: $AB,BC,CD,DA,EF,FG,GH,HE,AE,BF,CG$, and $DH$, where a twoletter string represents the set containing the letters

the following 4sets: $ABCD,EFGH,BCFG,AEDH,ABFE$, and $CDHG$, where the fourletter string represents the set containing the letters, and

$X$ itself
The map $\tau$ is defined as follows: all the singletons are mapped to vertex, all the doubletons to edge, the 4sets to face, and $X$ to cube. The following diagram is useful as a visual reference.
(14,3)(2,4) \pssetunit=2cm \pspolygon[linestyle=dashed, dash=4pt 4pt](4,0.5)(2,0.5)(2,1.5)(4,1.5) \pspolygon[linestyle=dashed, dash=4pt 4pt](5,1)(4,1.5)(4,0.5)(5,1) \pspolygon(5,1)(4,1.5)(2,1.5)(3,1) \pspolygon(2,0.5)(2,1.5)(3,1)(3,1) \pspolygon(5,1)(3,1)(3,1)(5,1) \uput[l](5,1)B \uput[d](3,1)D \uput[u](3,1)C \uput[l](5,1)A \uput[d](4,0.5)E \uput[r](2,0.5)H \uput[u](2,1.5)G \uput[u](4,1.5)F
Finally, $\#$ is defined as the inclusion relation: $P\#Q$ iff $P\subseteq Q$ or $Q\subseteq P$. Then $(\Gamma,\#,T,\tau)$ is a BuekenhoutTits geometry.
BuekenhoutTits geometries are generalizations of projective and affine geometries, and indeed incidence geometries in general. They also include examples where geometric meanings may not be apparent at first sight. Below are two such examples:

A bipartite graph can be thought of as a BuekenhoutTits geometry: we can take $\Gamma$ as the set of vertices, and two vertices are of the same type if they have the same chromatic number, and are incident if they are either the same, or there is an edge connecting them.

Let $G$ be a group and $\{G_{i}\mid i\in I\}$ be a set of subgroups of $G$, indexed by some set $I$. Let $\Gamma$ be the set of all (left) cosets of $G_{i}$, for all $i\in I$. Note that $xG_{i}=yG_{j}$ iff $i=j$ and $y^{{1}}x\in G_{i}$. Call $xG_{i}$ and $yG_{j}$ to be of the same type if $i=j$, and $xG_{i}\#yG_{j}$ if $xG_{i}\cap yG_{j}\neq\varnothing$. Then the geometry defined is a BuekenhoutTits geometry.
An incidence structure can be thought of as a BuekenhoutTits geometry of rank 2.
Remark. The graph of a geometry $(\Gamma,\#,T,\tau)$ is the pair $(\Gamma,\#)$, where objects are vertices of the graph, and $(a,b)$ is an edge of the graph iff $a\#b$. Properly speaking, the associated graph is a pseudograph since $\#$ is reflexive, so that there is a loop for each vertex.
References
 1 Handbook of Incidence Geometry, edited by Francis Buekenhout, Elsevier Science Publishing Co. (1995)
 2 M. Aschbacher, Finite Group Theory, Cambridge University Press (2000)
 3 P. J. Cameron, Projective and Polar Spaces, QMW Maths Notes, 13, London: Queen Mary and Westfield College School of Mathematical Sciences (2000)
Mathematics Subject Classification
51E24 no label found20C33 no label found05B25 no label found Forums
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