A Veblen-Wedderburn system is also called a quasifield.
Usually, we write instead of .
For any , by defining a ternary operation on , given by
it is not hard to see that is a ternary ring. In fact, it is a linear ternary ring because and .
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 be a Veblen-Wedderburn system. If the multiplication , in addition to be right distributive over , is also left distributive over , then is a semifield. If , on the other hand, is associative, then is an abelian nearfield (a nearfield such that is commutative).
- 1 R. Casse, Projective Geometry, An Introduction, Oxford University Press (2006)
|Date of creation||2013-03-22 19:15:06|
|Last modified on||2013-03-22 19:15:06|
|Last modified by||CWoo (3771)|