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Translation planes
Let V be a vector space over a skewfield K. A spread in V is a set of subspaces of V such that the following three properties hold.

Spreads are coordinatized by spreadsets. In fact, given a spreadset in a vector space V, the set

is a spread in the direct product

Conversely, given a spread in a vector space V, choose three pairwise distinct elements Then and denote the corresponding projections by It turns out that the set

is a spreadset in W, and

is an isomorphism mapping the spread onto the original spread .

Each spread defines a translation plane.

Theorem. Let be a spread in a vector space V. The set of lines is defined to be

Then is a translation plane.

Conversely, each translation plane is isomorphic to a translation plane derived from some spread. The notion of the kernel of a spreadset implies the concept of the kernel of a spread. More precisely,

Theorem. Let be a spread in a vector space V. Then the kernel of is defined to be

The kernel is a skewfield, V is a vector space over the kernel and our spread becomes a spread in the -vector space V. Finally, the original skewfield K is a subskewfield of the kernel.

See also

Contributed by Hauke Klein
Version $Id: spread.html,v 1.2 2001/07/23 10:16:59 hauke Exp $