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A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall [1] 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 ...
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A . In computer science , a ternary operator is an operator that takes three arguments as input and returns one output.
It is called a shear plane if any (or all) of its ternary rings are left quasifields. The plane does not uniquely determine the ring; all 4 nonabelian quasifields of order 9 are ternary rings for the unique non-Desarguesian translation plane of order 9. These differ in the fundamental quadrilateral used to construct the plane (see Weibel 2007).
Every projective plane can be coordinatized by at least one planar ternary ring. [5] For translation planes, it is always possible to coordinatize with a quasifield. [6] However, some quasifields satisfy additional algebraic properties, and the corresponding planar ternary rings coordinatize translation planes which admit additional symmetries.
The ternary operator is linear if T(x, m, k) = x⋅m + k. When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring. Algebraic properties of this planar ternary coordinate ring turn out to correspond to ...
If this property holds in the affine plane defined by a ternary ring, then there is an equivalence relation between "vectors" defined by pairs of points from the plane. [14] Furthermore, the vectors form an abelian group under addition; the ternary ring is linear and satisfies right distributivity: (+) = +.
Other classification schemes exist. One of the simplest is based on special types of planar ternary ring (PTR) that can be used to coordinatize the projective plane. These types are fields, skewfields, alternative division rings, semifields, nearfields, right nearfields, quasifields and right quasifields. [8]
Division ring – a ring in which every non-zero element has a multiplicative inverse; Semigroup – an algebraic structure consisting of a set together with an associative binary operation; Monoid – a semigroup with an identity element; Planar ternary ring – has an additive and multiplicative loop structure; Problems in loop theory and ...