A quasifield is a structure, where + and . binary operations on Q, satisfying these axioms :
Strictly speaking, this is the definition of a left quasifield. A right quasifield is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a semifield, in the sense in which the term is used in projective geometry.
Although not assumed, one can prove that the axioms imply that the additive group is abelian.
The kernel K of a quasifield Q is the set of all elements c such that :
Restricting the binary operations + and . to K, one can shown that (K,+,.) is a division ring .
One can now make a vector space of Q over K, with the following scalar multiplication :
As the order of any finite division ring is a prime power, this means that the order of any finite quasifield is also a prime power.
Given a quasifield , we define a ternary map by
One can then verify that satisfies the axioms of a planar ternary ring. Associated to is its corresponding projective plane. The projective planes constructed this way are characterized as follows: a projective plane is a translation plane with respect to the line at infinity if and only if its associated planar ternary ring is a quasifield.