arvau

Wikipedia / Planar ternary ring

In mathematics, an algebraic structure may be called a ternary system. A planar ternary ring (PTR) or ternary field is a special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field. 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.

There is wide variation in the terminology. 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. The term "ternary ring" often means a planar ternary ring, but it can also simply mean a ternary system.

Definition

A planar ternary ring is a mapping which satisfies these five axioms:

  1. ;

  2. ;

  3. ;

  4. ; and

  5. .

is finite, the third and fifth axioms are equivalent in the presence of the fourth.

still satisfies the first two axioms.

Binary operations

Addition

. is a loop with identity element 0.

Multiplication

is also a loop, with identity element 1.

Linear PTR

is said to be linear. For example, the planar ternary ring associated to a quasifield is (by construction) linear.

Connection with projective planes

, one can construct a projective plane with point set P and line set L as follows:.)

Let

  • , and

  • .

, the incidence relation 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) and u = (0).) by y = T(x,a,b) if and only if the point (x,y) lies on the line which joins (a) with (0,b). 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.

Intuition

The algebraic structure of a planar ternary ring (PTR) corresponds directly to the coordinates of an affine plane. While PTRs are often used to classify projective planes, the connection is best understood by examining the affine case first.

In affine geometry, points are described using Cartesian coordinates. This method is applicable even in non-Desarguesian geometries; in such contexts, the coordinate components always obey the structure of a PTR. By contrast, the homogeneous coordinates typically used in projective geometry are unavailable in non-Desarguesian contexts. Thus, the simplest analytic way to construct a projective plane in this context is to start with an affine plane and extend it by adding a "line at infinity".

In an affine plane, the algebraic representation of lines depends on the geometry:

.

We can now derive the analytic) by including the line at infinity. Formally, the projective plane is described as:

denotes the line at infinity.

To fully define the geometry, the line at infinity is further decomposed:

) not lying on that affine line.

Related algebraic structures

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 .

A linear PTR whose additive loop is associative (and thus a group), is called a cartesian group. In a cartesian group, the mappings

. Since cartesian groups are groups under addition, we revert to using a simple "+" for the additive operation.

A quasifield. Addition in any quasifield is commutative.

A semifield

A planar nearfield is a quasifield whose multiplicative loop is associative (and hence a group). Not all nearfields are planar nearfields.

Notes

References