Planar ternary ring

Definition

A planar ternary ring is a structure ( R , T ) where R is a set containing at least two distinct elements, called 0 and 1, and T : R 3 → R is a mapping which satisfies these five axioms:

1. T ( a , 0 , b ) = T ( 0 , a , b ) = b , ∀ a , b ∈ R ;
2. T ( 1 , a , 0 ) = T ( a , 1 , 0 ) = a , ∀ a ∈ R ;
3. ∀ a , b , c , d ∈ R , a ≠ c , there is a unique x ∈ R such that : T ( x , a , b ) = T ( x , c , d ) ;
4. ∀ a , b , c ∈ R , there is a unique x ∈ R , such that T ( a , b , x ) = c ; and
5. ∀ a , b , c , d ∈ R , a ≠ c , the equations T ( a , x , y ) = b , T ( c , x , y ) = d have a unique solution ( x , y ) ∈ R 2 .

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

No other pair (0', 1') in R 2 can be found such that T still satisfies the first two axioms.

Addition

Define a ⊕ b = T ( a , 1 , b ) . The structure ( R , ⊕ ) is a loop with identity element 0.

Multiplication

Define a ⊗ b = T ( a , b , 0 ) . The set R 0 = R ∖ { 0 } is closed under this multiplication. The structure ( R 0 , ⊗ ) is also a loop, with identity element 1.

Linear PTR

A planar ternary ring ( R , T ) is said to be linear if T ( a , b , c ) = ( a ⊗ b ) ⊕ c , ∀ a , b , c ∈ R . For example, the planar ternary ring associated to a quasifield is (by construction) linear.

Connection with projective planes

Given a planar ternary ring ( R , T ) , one can construct a projective plane with point set P and line set L as follows: (Note that ∞ is an extra symbol not in R .)

Let

Then define, ∀ a , b , c , d ∈ R , the incidence relation I 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). The ternary operation is now defined on the coordinate symbols (except ∞ ) 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.

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 Dembowski (1968, p. 129).

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

x ⟶ − x ⊗ a + x ⊗ b , and x ⟶ a ⊗ x − b ⊗ x

must be permutations whenever a ≠ b . Since cartesian groups are groups under addition, we revert to using a simple "+" for the additive operation.

A quasifield is a cartesian group satisfying the right distributive law: ( x + y ) ⊗ z = x ⊗ z + y ⊗ z . Addition in any quasifield is commutative.

A semifield is a quasifield which also satisfies the left distributive law: x ⊗ ( y + z ) = x ⊗ y + x ⊗ z .

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