Planar ternary ring
A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall[1] to construct projective planes by means of coordinates.
Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication.
Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here.
A planar ternary ring is a structure
is a set containing at least two distinct elements, called 0 and 1, and
is finite, the third and fifth axioms are equivalent in the presence of the fourth.
For example, the planar ternary ring associated to a quasifield is (by construction) linear.
in this way: Every projective plane can be constructed in this way, starting with an appropriate planar ternary ring.
However, two nonisomorphic planar ternary rings can lead to the construction of isomorphic projective planes.
Conversely, given any projective plane π, by choosing four points, labelled o, e, u, and v, no three of which lie on the same line, coordinates can be introduced in π so that these special points are given the coordinates: o = (0,0), e = (1,1), v = (
[7] The ternary operation is now defined on the coordinate symbols (except
The axioms defining a projective plane are used to show that this gives a planar ternary ring.
Linearity of the PTR is equivalent to a geometric condition holding in the associated projective plane.
[8] The connection between planar ternary rings (PTRs) and two-dimensional geometries, specifically projective and affine geometries, is best described by examining the affine case first.
In affine geometry, points on a plane are described using Cartesian coordinates, a method that is applicable even in non-Desarguesian geometries — there, coordinate-components can always be shown to obey the structure of a PTR.
By contrast, homogeneous coordinates, typically used in projective geometry, are unavailable in non-Desarguesian contexts.
Thus, the simplest analytic way to construct a projective plane is to start with an affine plane and extend it by adding a "line at infinity"; this bypasses homogeneous coordinates.
This representation extends to non-Desarguesian planes through the ternary operation of a PTR, allowing a line to be expressed as
Lines parallel to the y-axis are expressed by
We now show how to derive the analytic representation of a general projective plane given at the start of this section.
To do so, we pass from the affine plane, represented as
represents an affine plane in Cartesian coordinates and includes all finite points, while
is an affine line which we give its own Cartesian coordinate system, and
consists of a single point not lying on that affine line, which we represent using the symbol
PTR's which satisfy additional algebraic conditions are given other names.
These names are not uniformly applied in the literature.
The following listing of names and properties is taken from Dembowski (1968, p. 129).
A quasifield is a cartesian group satisfying the right distributive law:
A semifield is a quasifield which also satisfies the left distributive law:
A planar nearfield is a quasifield whose multiplicative loop is associative (and hence a group).
