Planar ternary ring

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In mathematics, a planar ternary ring (PTR) or ternary field is an algebraic structure (R,T), where R is a non-empty set, and T \colon R^3 \to R is a mapping satisfying certain axioms. A planar ternary ring is not a ring in the traditional sense. Planar ternary rings are of importance in the study of projective planes.

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[edit] Definition

A planar ternary ring is a structure (R,T) where R is a nonempty set, containing distinct elements called 0 and 1, and T\colon R^3\to R satisfies these five axioms:

  1. T(a,0,b)=T(0,a,b)=b\quad \forall a,b \in R;
  2. T(1,a,0)=T(a,1,0)=a\quad \forall a \in R;
  3. \forall a,b,c,d \in R, a\neq c, there is a unique x\in R such that : T(x,a,b) = T(x,c,d);
  4. \forall a,b,c \in R, there is a unique x \in R, such that T(a,b,x) = c; and
  5. \forall a,b,c,d \in R, a\neq c, the equations T(a,x,y) = b,T(c,x,y) = d have a unique solution (x,y)\in 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 R2 can be found such that T still satisfies the first two axioms.

[edit] Binary operations

[edit] Addition

Define a\oplus b=T(1,a,b). The structure (R,\oplus) turns out be a loop with identity element 0.

[edit] Multiplication

Define a\otimes b=T(a,b,0). The set R_{0} = R\backslash \{0\} turns out be closed under this multiplication. The structure (R_{0},\otimes) also turns out to be a loop with identity element 1.

[edit] Linear PTR

A planar ternary ring (R,T) is said to be linear if T(a,b,c)=(a\otimes b)\oplus c\quad \forall a,b,c \in R. For example, the planar ternary ring associated to a quasifield is (by construction) linear.

[edit] Connection with projective planes

Given a planar ternary ring (R,T), one can construct a projective plane in this way (\infty is a random symbol not in R):

 ((a,b),[c,d])\in I \Longleftrightarrow T(c,a,b)=d
 ((a,b),[c])\in I \Longleftrightarrow a=c
  ((a,b),[\infty])\notin I
 ((a), [c,d])\in I \Longleftrightarrow a=c
 ((a), [c])\notin I 
 ((a),[\infty])\in I  
 (((\infty),[c,d])\notin I
 ((\infty),[a])\in I 
 ((\infty),[\infty])\in I

One can prove that every projective plane is constructed in this way starting with a certain planar ternary ring. However, two nonisomorphic planar ternary rings can lead to the construction of isomorphic projective planes.