### Related pages

Homologies and elations of P(K,T)
Translation planes
A quasifield is a right distributive cartesian group. In more concrete terms, a quasifield consists of a set K and two binary operations such that the following axioms hold.

• (K,+) is a group. Let 0 be it's neutral element.
• We have 0x=0 for all elements x of K, and there exists an element 1 of K, distinct from 0 with 1x=x1=x for each x in K.
• .
• .
• .
It turns out that a quasifield is a vector space over some skewfield called the kernel of the quasifield.

Theorem. Let K be a quasifield. Then the additive group (K,+) is abelian and

is a subskewfield of K, called the kernel of K. Moreover, using the multiplication in K, the quasifield K becomes a vector space over it's kernel.

Since a quasifield 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 translation planes.

Theorem. A ternary field (K,T) is a quasifield if and only if the line at infinity of the projective plane P(K,T) is a translation axis.

The kernel of a quasifield also has a simple geometrical interpretation, which is easily derived from the explicit computation of the homologies and elations of P(K,T).

Theorem. Let K be a quasifield and write Then

.