Quasifield

In mathematics, a quasifield is an algebraic structure (Q,+,.) where + and . are binary operations on Q, much like a division ring, but with some weaker conditions.

Definition

A quasifield $(Q,+,.)$ is a structure, where + and . binary operations on Q, satisfying these axioms :

* $(Q,+)$ is a group
* $(Q\_\{0\},.)$ is a loop, where
* $a.(b+c)=a.b+a.c\; forall\; a,b,c\; in\; Q$ (left distributivity)
* $a.x=b.x+c$ has exactly one solution $forall\; a,b,c\; in\; Q$

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 $(Q,+)$ is abelian.

Kernel

The kernel K of a quasifield Q is the set of all elements c such that :
* $a.(b.c)=(a.b).c\; forall\; a,bin\; Q$
* $(a+b).c=(a.c)+(b.c)\; forall\; a,bin\; Q$

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 :$v\; otimes\; l\; =\; v\; .\; l\; forall\; vin\; Q,lin\; K$

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.

Projective planes

Given a quasifield $Q$, we define a ternary map $Tcolon\; Q\; imes\; Q\; imes\; Q\; o\; Q$ by

$T(a,b,c)=a.b+c\; forall\; a,b,cin\; Q$

One can then verify that $(Q,T)$ satisfies the axioms of a planar ternary ring. Associated to $(Q,T)$ 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.

References