Abstract
A d-net is a connected semilinear incidence structure π such that (D1) every plane is a net, (D2) the intersection of two subspaces is connected, (D3) if two planes in a 3-space have a point in common then they have a second point in common, and (D4) the minimum number of points which generate π is d . Let V be a vector space over a skew field F , and W a subspace of finite codimension d . Let P , L be the set of d -, ( d - 1)-dimensional subspaces respectively of V whose intersection with W is the zero vector. The incidence structure ( P , L ⊇ ) is called an attenuated space . We show every d -net for finite d ⩾ 3 is an attenuated space. We also characterize d -nets (together with AG ( d , 2)) as those incidence structures belonging to the diagram where signifies a projective plane and signifies a net.
Published Version
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