Related pagesTranslation planes
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, la denotes the left multiplication by the element a, i.e. la(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