Quasifield

In mathematics, a quasifield is an algebraic structure formula_1 where + and formula_2 are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition.
A quasifield formula_1 is a structure, where + and formula_4 are 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 formula_11 is abelian. Thus, when referring to an "abelian quasifield", one means that formula_12 is abelian.
Kernel.
The kernel K of a quasifield Q is the set of all elements c such that :
Restricting the binary operations + and formula_2 to K, one can shown that formula_16 is a division ring.
One can now make a vector space of Q over K, with the following scalar multiplication :
formula_17
As a finite division ring is a finite field by Wedderburn's theorem, the order of the kernel of a finite quasifield is a prime power. The vector space construction implies that the order of any finite quasifield must also be a prime power.
Examples.
All division rings, and thus all fields, are quasifields. 
The smallest quasifields are abelian and unique. They are the finite fields of orders up to and including eight. The smallest quasifields which are not division rings are the four non-abelian quasifields of order nine; they are presented in and .
Projective planes.
Given a quasifield formula_18, we define a ternary map formula_19 by
formula_20
One can then verify that formula_21 satisfies the axioms of a planar ternary ring. Associated to formula_21 is its corresponding projective plane. The projective planes constructed this way are characterized as follows;
the details of this relationship are given in . 
A projective plane is a translation plane with respect to the line at infinity if and only if any (or all) of its associated planar ternary rings are right quasifields. It is called a "shear plane" if any (or all) of its ternary rings are left quasifields.
The plane does not uniquely determine the ring; all 4 nonabelian quasifields of order 9 are ternary rings for the unique non-Desarguesian translation plane of order 9. These differ in the fundamental quadrilateral used to construct the plane (see Weibel 2007).
History.
Quasifields were called "Veblen-Wedderburn systems" in the literature before 1975, since they were first studied in the
1907 paper (Veblen-Wedderburn 1907) by O. Veblen and J. Wedderburn. Surveys of quasifields and their applications to projective planes may be found in and .