Related pagesTranslation planes
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.