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Concept
Planar ternary ring
Summary
In mathematics, an algebraic structure consisting of a non-empty set and a ternary mapping may be called a ternary system. A planar ternary ring (PTR) or ternary field is 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 is defined by . 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.
A planar ternary ring is a structure where is a set containing at least two distinct elements, called 0 and 1, and is a mapping which satisfies these five axioms:
there is a unique such that : ;
there is a unique , such that ; and
the equations have a unique solution .
When is finite, the third and fifth axioms are equivalent in the presence of the fourth.
No other pair (0', 1') in can be found such that still satisfies the first two axioms.
Define . The structure is a loop with identity element 0.
Define . The set is closed under this multiplication. The structure is also a loop, with identity element 1.
A planar ternary ring is said to be linear if .
For example, the planar ternary ring associated to a quasifield is (by construction) linear.
Given a planar ternary ring , one can construct a projective plane with point set P and line set L as follows: (Note that is an extra symbol not in .)
Let
and
Then define, , the incidence relation in this way:
Every projective plane can be constructed in this way, starting with an appropriate planar ternary ring. However, two nonisomorphic planar ternary rings can lead to the construction of isomorphic projective planes.
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