Planar ternary ring

Summary

A planar ternary ring is a set with a ternary operation $T(x, m, b)$ that plays the role of $x m + b$ in the equation $y = x m + b$ of a line. Marshall Hall showed that every projective plane can be coordinatized by a planar ternary ring. Conversely, every planar ternary ring gives a projective plane. Algebraic properties of the ternary ring match geometric properties of the plane. For instance, the plane is a Desarguesian plane when the ring is a division ring with $T(x, m, b) = x m + b$. It is the main tool for constructing and classifying non-Desarguesian planes.