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# 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 $\left(Q,+,.\right)$ is a structure, where + and . binary operations on Q, satisfying these axioms :

• $\left(Q,+\right)$ is a group
• $\left(Q_\left\{0\right\},.\right)$ is a loop, where $Q_\left\{0\right\} = Q backslash \left\{0\right\}$
• $a.\left(b+c\right)=a.b+a.c forall a,b,c in Q$ (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 $\left(Q,+\right)$ is abelian.

## Kernel

The kernel K of a quasifield Q is the set of all elements c such that :

• $a.\left(b.c\right)=\left(a.b\right).c forall a,bin Q$
• $\left(a+b\right).c=\left(a.c\right)+\left(b.c\right) forall a,bin Q$

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 $Tcolon Qtimes Qtimes Qto Q$ by

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

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

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