---
tag: E6F61940F9
kind: definition
name: Planar ternary ring
msc: [51A25, 51A35]
defines: Q0NXXFADT9
uses: [WC47MFR6GW]
sources:
  - url: https://en.wikipedia.org/wiki/Planar_ternary_ring
---

## Definition

A *planar ternary ring* is a set $R$ with two distinct elements $0$ and $1$ and a [[ternary operation|WC47MFR6GW]] $T \colon R^3 \to R$ such that:

1. $T(a, 0, c) = T(0, b, c) = c$ for all $a, b, c \in R$;
2. $T(1, a, 0) = T(a, 1, 0) = a$ for all $a \in R$;
3. for all $a, b, c, d \in R$ with $a \ne c$, there is a unique $x \in R$ with $T(x, a, b) = T(x, c, d)$;
4. for all $a, b, c \in R$, there is a unique $x \in R$ with $T(a, b, x) = c$;
5. for all $a, b, c, d \in R$ with $a \ne c$, there is a unique pair $(x, y) \in R^2$ with $T(a, x, y) = b$ and $T(c, x, y) = d$.

## Remarks

The intended reading is $T(x, m, b) = x m + b$.
Clause 3 says that two lines of different slopes meet in exactly one point.
Clause 4 says that there is one line of given slope through a given point.
Clause 5 says that two points with different first coordinates lie on exactly one non-vertical line.

Example: a field $k$ with $T(a, b, c) = a b + c$ is a planar ternary ring.
Check: clauses 1 and 2 are $a \cdot 0 + c = 0 \cdot b + c = c$ and $1 \cdot a = a \cdot 1 = a$.
Clause 3 is $x(a - c) = d - b$, solved by $x = (d - b)(a - c)^{-1}$, and clause 4 is solved by $x = c - a b$.
Clause 5 is a $2 \times 2$ linear system with determinant $a - c \ne 0$.

Non-example: $\Z$ with $T(a, b, c) = a b + c$ fails clause 3.
For $a = 2$, $b = 0$, $c = 0$, $d = 1$ the equation $2x = 1$ has no solution in $\Z$.

A planar ternary ring has an addition $a + b = T(a, 1, b)$ and a multiplication $a \cdot b = T(a, b, 0)$.
It is *linear* when $T(a, b, c) = a \cdot b + c$ for all $a, b, c$.
Choosing four points in general position in a projective plane gives a planar ternary ring that coordinatizes it, and every planar ternary ring arises this way.