Related pagesTranslation planes
These sets are called the left-, middle- and right nucleus of K, respectively. The intersection
is called the nucleus of K, and finally
is the center of K. Of course, the right nucleus is just the kernel of the quasifield K.
It turns out that the center Z(K) is a subfield of K, and K becomes a division algebra over this field.
Since a semifield K is a cartesian group, we may think of K as a ternary field by using the ternary operation T(s,x,t)=sx+t. In particular, we have a projective plane P(K) over K. These planes are exactly the projective planes of Lenz type at least V.
Theorem. A ternary field (K,T) is a semifield if and only if the line at infinity of the projective plane P(K,T) is a translation axis and the point is a dual translation axis, respectively.