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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.
## Definition

## Kernel

## Projective planes

## References

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 $Q\_\{0\}\; =\; Q\; backslash\; \{0\}$
- $a.(b+c)=a.b+a.c\; forall\; a,b,c\; in\; Q$ (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.

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.

Given a quasifield $Q$, we define a ternary map $Tcolon\; Qtimes\; Qtimes\; Qto\; Q$ 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.

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Last updated on Monday June 23, 2008 at 11:56:48 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday June 23, 2008 at 11:56:48 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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