In mathematics, an algebraic structure ( R , T ) {\displaystyle (R,T)} consisting of a non-empty set R {\displaystyle R} and a ternary mapping T : R 3 → R {\displaystyle T\colon R^{3}\to R\,} may be called a ternary system. A planar ternary ring (PTR) or ternary field is a special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation T {\displaystyle T} is defined by T ( a , b , c ) = a b + c {\displaystyle T(a,b,c)=ab+c} . Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication. There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "ternary ring" often means a planar ternary ring, but it can also simply mean a ternary system.
Definition A planar ternary ring is a structure ( R , T ) {\displaystyle (R,T)} where R {\displaystyle R} is a set containing at least two distinct elements, called 0 and 1, and T : R 3 → R {\displaystyle T\colon R^{3}\to R\,} is a mapping which satisfies these five axioms:
T ( a , 0 , b ) = T ( 0 , a , b ) = b , ∀ a , b ∈ R {\displaystyle T(a,0,b)=T(0,a,b)=b,\quad \forall a,b\in R} ;
T ( 1 , a , 0 ) = T ( a , 1 , 0 ) = a , ∀ a ∈ R {\displaystyle T(1,a,0)=T(a,1,0)=a,\quad \forall a\in R} ;
∀ a , b , c , d ∈ R , a ≠ c {\displaystyle \forall a,b,c,d\in R,a\neq c} , there is a unique x ∈ R {\displaystyle x\in R} such that : T ( x , a , b ) = T ( x , c , d ) {\displaystyle T(x,a,b)=T(x,c,d)\,} ;
∀ a , b , c ∈ R {\displaystyle \forall a,b,c\in R} , there is a unique x ∈ R {\displaystyle x\in R} , such that T ( a , b , x ) = c {\displaystyle T(a,b,x)=c\,} ; and
∀ a , b , c , d ∈ R , a ≠ c {\displaystyle \forall a,b,c,d\in R,a\neq c} , the equations T ( a , x , y ) = b , T ( c , x , y ) = d {\displaystyle T(a,x,y)=b,T(c,x,y)=d\,} have a unique solution ( x , y ) ∈ R 2 {\displaystyle (x,y)\in R^{2}} . When R {\displaystyle R} is finite, the third and fifth axioms are equivalent in the presence of the fourth. No other pair (0', 1') in R 2 {\displaystyle R^{2}} can be found such that T {\displaystyle T} still satisfies the first two axioms.
Binary operations
Addition Define a ⊕ b = T ( a , 1 , b ) {\displaystyle a\oplus b=T(a,1,b)} . The structure ( R , ⊕ ) {\displaystyle (R,\oplus )} is a loop with identity element 0.
Multiplication Define a ⊗ b = T ( a , b , 0 ) {\displaystyle a\otimes b=T(a,b,0)} . The set R 0 = R ∖ { 0 } {\displaystyle R_{0}=R\setminus \{0\}\,} is closed under this multiplication. The structure ( R 0 , ⊗ ) {\displaystyle (R_{0},\otimes )} is also a loop, with identity element 1.
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