Planar ternary ring

In mathematics, a ternary ring is an algebraic structure $(R,T)$ consisting of a non-empty set $R$ and a ternary mapping $T \colon R^3 \to R \,$, and a planar ternary ring (PTR) or ternary field is special sort of a ternary ring used by Hall (1943) to give coordinates to projective planes. A planar ternary ring is not a ring in the traditional sense.

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 $R^2$ can be found such that $T$ still satisfies the first two axioms.

Binary operations

Addition

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

Multiplication

Define $a\otimes b=T(a,b,0)$. The set $R_{0} = R \setminus \{0\} \,$ 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 $T(a,b,c)=(a\otimes b)\oplus c\quad \forall a,b,c \in R$. For example, the planar ternary ring associated to a quasifield is (by construction) linear.

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 ($\forall a,b,c,d \in R$):

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

$((a,b),[c])\in I \Longleftrightarrow a=c$

$((a,b),[\infty])\notin I$

$((a), [c,d])\in I \Longleftrightarrow a=c$

$((a), [c])\notin I$

$((a),[\infty])\in I$

$(((\infty),[c,d])\notin I$

$((\infty),[a])\in I$

$((\infty),[\infty])\in I$

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.

References

• Albert, A. Adrian; Sandler, Reuben (1968). An Introduction to Finite Projective Planes. New York: Holt, Rinehart and Winston. 
• Hall, Marshall (1943), "Projective planes", Transactions of the American Mathematical Society (American Mathematical Society) 54 (2): 229–277, ISSN 0002-9947, JSTOR 1990331, MR0008892