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 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.