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  "body": "\n## Definition\n\nLet $I$ be a finite set, the set of types.\nLet $X$ be a set with a reflexive and symmetric relation $*$, called incidence.\nLet $\\tau \\colon X \\to I$ be a map, called the type function.\nThe triple $(X, *, \\tau)$ is a *Buekenhout geometry* (or *geometry*) over $I$ if:\n\n1. if $x * y$ and $\\tau(x) = \\tau(y)$, then $x = y$;\n2. every flag, that is, every set of pairwise incident elements of $X$, is contained in a flag $C$ with $\\tau(C) = I$.\n\n## Remarks\n\nA flag $C$ with $\\tau(C) = I$ is a *chamber*; by clause 1 it has exactly one element of each type.\nThe number $\\abs{I}$ is the rank.\nA geometry of rank $2$ is the same as an [[incidence structure|FGKNAFNHA3]] whose flags extend, such as a projective plane or a [[generalized polygon|PCT2HY9AAZ]].\nThe 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 $F$.\nThe *Buekenhout diagram* records, for each pair of types $i \\ne j$, the class of the rank $2$ residues of type $\\set{i, j}$.\nTits buildings give geometries whose diagrams are Coxeter diagrams.\n\nExample: let $\\mathcal{P}$ be an $n$-dimensional [[projective space|ZT9A6JK8QF]].\nTake $X$ 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 $y \\subseteq x$.\nClause 1 holds since two distinct subspaces of the same dimension are not contained in each other.\nClause 2 holds since every chain of subspaces can be refined to a chain with one subspace of each dimension.\n\nNon-example: take the lines of a projective plane, all of one type, with $x * y$ meaning that $x$ and $y$ meet.\nAny two lines meet, so two distinct incident elements have the same type and clause 1 fails.\n",
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