## Related pagesTranslation planesQuasifields Spreadsets |

- Given two distinct element then U+W=V.
- Each v in V distinct from zero is contained in exactly one element of .

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.

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