Quasifield
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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.
[edit] Definition
A quasifield (Q, + ,.) is a structure, where + and . binary operations on Q, satisfying these axioms :
- (Q, + ) is a group
- (Q0,.) is a loop, where
(left distributivity)- a.x = b.x + c has exactly one solution
Strictly speaking, this is the definition of a left quasifield. A right quasifield is similarly defined, but satisfies right distributivity instead.
Although not assumed, one can prove that the axioms imply that the additive group (Q, + ) is abelian.
[edit] Kernel
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.
[edit] Projective planes
Given a quasifield Q, we define a ternary map 
by
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.
