Planar ternary ring

From Wikipedia, the free encyclopedia

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 crucial importance in the study of projective planes.

1 Definition
2 Binary operations
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 $T\colon R^3\to R$ satisfies these five axioms:

$T(a,0,b)=T(0,a,b)=b\quad \forall a,b \in R$ ;
$T(1,a,0)=T(a,1,0)=a\quad \forall a \in R$ ;
$\forall a,b,c,d \in R, a\neq c$ , there is a unique $x\in R$ such that : $T (x, a, b) = T (x, c, d)$ ;
$\forall a,b,c \in R$ , there is a unique $x \in R$ , such that $T (a, b, x) = c$ ; and
$\forall a,b,c,d \in R, a\neq c$ , 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 couple (0',1') in $R 2$ 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.

[edit] Multiplication

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

[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 $R$ ):

$P=\{(a,b)|a,b\in R\}\cup \{(a)|a \in R \}\cup \{(\infty)\}$
$B=\{[a,b]|a,b \in R\}\cup\{[a]|a \in R \}\cup \{[\infty]\}$
We define the incidence relation $I$ in this way ( $\forall a,b,c,d \in R$ ):

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.

Retrieved from "http://en.wikipedia.org../../../p/l/a/Planar_ternary_ring.html"

Category: Abstract algebra