Planar ternary ring

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.