Polar space

rdf:langString Polar space

rdf:langString In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms: * Every subspace is isomorphic to a projective geometry Pd(K) with −1 ≤ d ≤ (n − 1) and K a division ring. By definition, for each subspace the corresponding d is its dimension. * The intersection of two subspaces is always a subspace. * For each point p not in a subspace A of dimension of n − 1, there is a unique subspace B of dimension n − 1 containing p and such that A ∩ B is (n − 2)-dimensional. The points in A ∩ B are exactly the points of A that are in a common subspace of dimension 1 with p. * There are at least two disjoint subspaces of dimension n − 1. It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (P,L), so that for each point p ∈ P andeach line l ∈ L, the set of points of l collinear to p, is either a singleton or the whole l. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.