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.
 (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.

.

.
Cartesian groups are characterized by the following geometrical property
of there projective planes.
Theorem. A ternary field is a
cartesian group if and only if the
projective plane over (K,T) is

transitive.
In particular, the projective planes over cartesian groups are exactly
the projective planes of
Lenz type at least II.
See also
Contributed by Hauke Klein
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