Ternary field
planar ternary ring
A set with two special elements,
and
, provided with a ternary operation
satisfying:
A) for all
;
B) for all
;
C) if ,
, then there is a unique
such that
;
D) if , then there is a unique
such that
;
E) if ,
, then there are unique
such that
and
.
Ternary fields were introduced in [a1] for the purpose of coordinatizing arbitrary, not necessarily Desarguesian, projective planes (cf. Desargues assumption; Desargues geometry; Projective plane). Slight variations of the original definition were given in [a2] and [a3], which is followed here. Given a projective plane, fix four points in general position: ,
,
,
, and let
,
and
. For the points
of
one chooses coordinates
with
running over a set
and
, assigning
to
and
to
. The projection of
from
on
is given coordinates
, and then
(see Fig.a1). The points on
get one coordinate
, with
, or
, where
is an extra symbol
, and the lines are coordinatized by
,
or
, as in Fig.a2.
Figure: t092430a
Figure: t092430b
The ternary operation on
is defined by
if and only if
lies on
. The properties A)–E) for
are then consequences of the axioms for a projective plane. Conversely, any ternary field coordinatizes a projective plane. It may happen that different ternary fields coordinatize the same plane, for a different choice of basis points
,
,
,
.
In case is finite, C) and D) are equivalent to D) and E); further, C) is then a consequence of D) and the existence of at most one
as in C).
On a ternary field , addition is defined by
; with this operation
is a loop with
as neutral element. Multiplication is defined by
; this makes
a loop with
as neutral element.
is said to be linear if
for all
,
,
. Linearity is equivalent to a very weak Desargues-type condition on triangles which are in perspective from the point
(cf. Configuration, in particular Desarguesian configuration, and also Desargues assumption). Other algebraic properties of
, such as associativity of addition or multiplication and left or right distributivity, can also be translated into certain Desargues-type conditions. In particular, a translation plane with
as translation line, i.e., a plane in which the group of
-translations is transitive on the points not on
, is coordinatized by a (left) quasi-field, which is a linear ternary field with associative addition satisfying the left distributive law
.
References
[a1] | M. Hall, "Projective planes" Trans. Amer. Math. Soc. , 54 (1943) pp. 229–277 |
[a2] | G. Pickert, "Projective Ebenen" , Springer (1975) |
[a3] | D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973) |
Ternary field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ternary_field&oldid=16138