---
tag: Q0NXXFADT9
kind: concept
name: Planar ternary ring
msc: [51A25, 51A35]
sources:
  - url: https://en.wikipedia.org/wiki/Planar_ternary_ring
same_as:
  wikidata: Q2405012
  wikipedia: Planar ternary ring
---

## Summary

A planar ternary ring is a set with a [[ternary operation|WC47MFR6GW]] $T(x, m, b)$ 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|DANFNN8C4Y]] 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|HPX85PA8NX]] 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|HYE5FT4M34]].

## Explanations

### L1 human

In the usual plane, a non-vertical line is the set of points with $y = x m + b$.
An abstract projective plane may have no multiplication and no addition.
Still, a rule tells which $y$ belongs to $x$ on the line with slope $m$ and intercept $b$.
That rule is the ternary operation $T(x, m, b)$.
Its axioms say exactly what is needed for two points to determine one line and two lines to meet once.