Planar ternary ring
From Wikipedia, the free encyclopedia
In mathematics, a ternary ring is an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping 
, 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.
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[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 satisfies these five axioms:
;
;
, there is a unique 
such that : T(x,a,b) = T(x,c,d);
, there is a unique , such that T(a,b,x) = c; and- , the equations T(a,x,y) = b,T(c,x,y) = d have a unique solution
.
When R is finite, the third and fifth axioms are equivalent in the presence of the fourth. No other pair (0',1') in R2 can be found such that T still satisfies the first two axioms.
[edit] Binary operations
[edit] Addition
Define 
. The structure 
turns out be a loop with identity element 0.
[edit] Multiplication
Define 
. The set 
turns out be closed under this multiplication. The structure 
also turns out to be a loop with identity element 1.
[edit] Linear PTR
A planar ternary ring (R,T) is said to be linear if 
. 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 (
is a random symbol not in R):
![B=\{[a,b]|a,b \in R\}\cup\{[a]|a \in R \}\cup \{[\infty]\}](http://upload.wikimedia.org/math/c/2/c/c2c09860adb0206229e680e9faf5b0cf.png)
- We define the incidence relation I in this way (
):
![((a,b),[c,d])\in I \Longleftrightarrow T(c,a,b)=d](http://upload.wikimedia.org/math/0/c/c/0cc2b05add45da264990a7b3bd207c70.png)
![((a,b),[c])\in I \Longleftrightarrow a=c](http://upload.wikimedia.org/math/9/d/e/9de9b4a17e55c7ca5efbad76206839a8.png)
![((a,b),[\infty])\notin I](http://upload.wikimedia.org/math/a/e/5/ae53fb4af48749dd76a390568d19df4b.png)
![((a), [c,d])\in I \Longleftrightarrow a=c](http://upload.wikimedia.org/math/9/5/f/95f186faa28bde1ae07426d1e5850a41.png)
![((a), [c])\notin I](http://upload.wikimedia.org/math/2/8/d/28da84db95499c50f6e9c1e82d03384c.png)
![((a),[\infty])\in I](http://upload.wikimedia.org/math/e/6/1/e61b295ffae96c2e02f944cf7dea03ea.png)
![(((\infty),[c,d])\notin I](http://upload.wikimedia.org/math/9/c/6/9c62a7566843822ece9b7e61dd0a390a.png)
![((\infty),[a])\in I](http://upload.wikimedia.org/math/a/9/d/a9d7a9e008c2fb1e8ec8e2631b69c143.png)
![((\infty),[\infty])\in I](http://upload.wikimedia.org/math/8/5/b/85b517722453203794164c609ee959be.png)
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.
[edit] References
- Hall, Marshall (1943), "Projective planes", Transactions of the American Mathematical Society 54: 229–277, MR0008892, ISSN 0002-9947, http://www.jstor.org/stable/1990331
