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 $a\ne b$, 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 $ab$ 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).