Veblen-Wedderburn system

A Veblen-Wedderburn system is an algebraic system over a set $R$ with two binary operations $+$ (called addition) and $\cdot$ (called multiplication) on $R$ such that

1.

there is a $0\in R$ , and that $R$ is an abelian group under $+$ , with $0$ the additive identity
2.

$R-\{0\}$ , together with $\cdot$ , is a loop (we denote $1$ as its identity element)
3.

$\cdot$ is right distributive over $+$ ; that is, $(a+b)\cdot c=a\cdot c+b\cdot c$
4.

if $a\neq b$ , then the equation $x\cdot a=x\cdot b+c$ has a unique solution in $x$

A Veblen-Wedderburn system is also called a quasifield.

Usually, we write $a b$ instead of $a\cdot b$ .

For any $a,b,c\in R$ , by defining a ternary operation $*$ on $R$ , given by

a*b*c:=ab+c,

it is not hard to see that $(R,*,0,1)$ is a ternary ring. In fact, it is a linear ternary ring because $ab=a*b*0$ and $a+c=a*1*c$ .

For example, any field, or more generally, any division ring, associative or not, is Veblen-Wedderburn. An example of a Veblen-Wedderburn system that is not a division ring is the Hall quasifield.

A well-known fact about Veblen-Wedderburn systems is that, the projective plane of a Veblen-Wedderburn system is a translation plane, and, conversely, every translation plane can be coordinatized by a Veblen-Wedderburn system. This is the reason why a translation plane is also called a Veblen-Wedderburn plane.

Remark. Let $R$ be a Veblen-Wedderburn system. If the multiplication $\cdot$ , in addition to be right distributive over $+$ , is also left distributive over $+$ , then $R$ is a semifield. If $\cdot$ , on the other hand, is associative, then $R$ is an abelian nearfield (a nearfield such that $+$ is commutative).

References

1 R. Casse, Projective Geometry, An Introduction, Oxford University Press (2006)

Title	Veblen-Wedderburn system
Canonical name	VeblenWedderburnSystem
Date of creation	2013-03-22 19:15:06
Last modified on	2013-03-22 19:15:06
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	9
Author	CWoo (3771)
Entry type	Definition
Classification	msc 51A35
Classification	msc 51A40
Classification	msc 51E15
Classification	msc 51A25
Synonym	Veblen-Wedderburn ring
Synonym	quasifield
Synonym	VW system
Synonym	V-W system