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. 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.

[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.