Veblen-Wedderburn system

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

  1. 1.

    there is a 0R, and that R is an abelian groupMathworldPlanetmath under +, with 0 the additive identity

  2. 2.

    R-{0}, together with , is a loop (we denote 1 as its identity elementMathworldPlanetmath)

  3. 3.

    is right distributivePlanetmathPlanetmath over +; that is, (a+b)c=ac+bc

  4. 4.

    if ab, then the equation xa=xb+c has a unique solution in x

A Veblen-Wedderburn system is also called a quasifield.

Usually, we write ab instead of ab.

For any a,b,cR, by defining a ternary operation * on R, given by


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 planeMathworldPlanetmath 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 , in addition to be right distributive over +, is also left distributive over +, then R is a semifield. If , on the other hand, is associative, then R is an abelian nearfield (a nearfield such that + is commutativePlanetmathPlanetmath).


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