A cartesian group is a
linear ternary field with
associative addition. In more concrete terms, a cartesian group
consists of a set K and two operations
such that the following axioms are satisfied.
Cartesian groups are characterized by the following geometrical property
of there projective planes.
- (K,+) is group. Let 0 be it's neutral element.
- We have 0x=x0=0 for all elements x of K, and there exists an element 1
of K, distinct from zero, such that 1x=x1=x for all x in K.
Theorem. A ternary field is a
cartesian group if and only if the
projective plane over (K,T) is
In particular, the projective planes over cartesian groups are exactly
the projective planes of
Lenz type at least II.
Contributed by Hauke Klein
Version $Id: cartgroup.html,v 1.1 2000/12/29 20:20:09 hauke Exp $