Definition
An n-dimensional building X is an abstract simplicial complex which is a union of subcomplexes A called apartments such that
- every k-simplex of X is within at least three n-simplices if k < n;
- any (n – 1 )-simplex in an apartment A lies in exactly two adjacent n-simplices of A and the graph of adjacent n-simplices is connected;
- any two simplices in X lie in some common apartment A;
- if two simplices both lie in apartments A and A ', then there is a simplicial isomorphism of A onto A ' fixing the vertices of the two simplices.
An n-simplex in A is called a chamber (originally chambre, i.e. room in French).
The rank of the building is defined to be n + 1.
Read more about this topic: Building (mathematics)
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