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Translation planes
A semifield is a left distributive quasifield. In more concrete terms, a semifield consists of a set K and two binary operations such that the following axioms hold.

• (K,+) is a group.
• There exists an element 1 of K distinct from zero with 1x=x1=x for each x in K.
• .
• .
• .
Given a semifield K, the three nuclei of K are defined to be the following sets.

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.

See also

Contributed by Hauke Klein
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