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 is a mapping satisfying certain axioms. A planar ternary ring is not a ring in the traditional sense. Planar ternary rings are of crucial importance in the study of projective planes.

Contents 1 Definition 2 Binary operations 2.1 Addition 2.2 Multiplication 2.3 Linear PTR 3 Connection with projective planes

[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 satisfies these five axioms:

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

When R is finite, the third and fifth axioms are equivalent in the presence of the fourth. No other couple (0',1') in R² can be found such that T still satisfies the first two axioms.

[edit] Binary operations

[edit] Addition

Define . The structure turns out be a loop.

[edit] Multiplication

Define . R₀ turns out be closed under this multiplication. The structure also turns out to be a loop.

[edit] Linear PTR

A planar ternary ring (R,T) is said to be linear if . 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 ( is a random symbol not in R):

• 
• 
• We define the incidence relation I in this way ():

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.