Planar ternary ring

In mathematics, an algebraic structure ${\displaystyle (R,T)}$ consisting of a non-empty set ${\displaystyle R}$ and a ternary mapping ${\displaystyle T\colon R^{3}\to R\,}$ may be called a ternary system. 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 ${\displaystyle T}$ is defined by ${\displaystyle T(a,b,c)=ab+c}$. 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.