## Related pagesTranslation planesSpreads Quasifields Finite spreads |

- .
- Given any two distinct element then the difference A-B is invertible.
- Given two vectors v,w in V with there exists an such that Av=w.

Of course, if the dimension of V is finite, the set is a spreadset if and only if and given any two vectors v,w in V, v distinct from zero, there is exactly one such that Av=w.

Spreadsets are almost the same objects as quasifields. In fact, the following simple theorem holds.

**Theorem.** Let K be a quasifield.
Then K becomes a vector space over its kernel
Ker(K) and

is a spreadset in this vector space. Here, l_{a} denotes the
left multiplication by the element a, i.e. l_{a}(x)=ax.

Conversely, given a spreadset in a vector space V, we choose an element e in V distinct from zero, and for each v in V there exists exactly one with . Using this notation, we define a multiplication on V by

Then V becomes a quasifield and the original spreadset is the set of left multiplications in this quasifield.

Note that the spreadset is an additive subgroup of End(V) if and only if K is a semifield. Moreover, is a subgroup of GL(V) if and only if K is a nearfield.

Using the construction in the theorem, the kernel of a quasifield becomes the centralizer of the corresponding spreadset, i.e.

Each spreadset yields a spread, and we get a translation plane, which will be written as .

**Theorem.** Given a spreadset
in a vector space V,
the translation plane
is desarguesian if and
only if

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