Planar ternary ring

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 o 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.

Definition

A planar ternary ring is a structure $$(R,T) where $$R is a nonempty set, containing distinct elements called 0 and 1, and $$Tcolon R^3 o R satisfies these five axioms:
# $$T(a,0,b)=T(0,a,b)=bquad forall a,b in R;
# $$T(1,a,0)=T(a,1,0)=aquad forall a in R;
# $$forall a,b,c,d in R, a eq c, there is a unique $$xin R such that : $$T(x,a,b)=T(x,c,d);
# $$forall a,b,c in R, there is a unique $$x in R, such that $$T(a,b,x)=c; and
# $$forall a,b,c,d in R, a eq c, the equations $$T(a,x,y)=b,T(c,x,y)=d have a unique solution $$(x,y)in R^2.

When $$R is finite, the third and fifth axioms are equivalent in the presence of the fourth.No other pair (0',1') in $$R^2 can be found such that $$T still satisfies the first two axioms.

Binary operations

Addition

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

Multiplication

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

Linear PTR

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

Connection with projective planes

Given a planar ternary ring $$(R,T), one can construct a projective plane in this way ($$infty is a random symbol not in $$R):
* $$P={(a,b)|a,bin R}cup {(a)|a in R }cup {(infty)}
* $$B={ [a,b] |a,b in R}cup{ [a] |a in R }cup { [infty] }
* We define the incidence relation $$I in this way ($$forall a,b,c,d in R): $$((a,b), [c,d] )in I Longleftrightarrow T(c,a,b)=d $$((a,b), [c] )in I Longleftrightarrow a=c $$ ((a,b), [infty] ) otin I $$((a), [c,d] )in I Longleftrightarrow a=c $$((a), [c] ) otin I $$((a), [infty] )in I $$(((infty), [c,d] ) otin 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.