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  "body": "\n## Definition\n\nA *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:\n\n1. $T(a, 0, c) = T(0, b, c) = c$ for all $a, b, c \\in R$;\n2. $T(1, a, 0) = T(a, 1, 0) = a$ for all $a \\in R$;\n3. for all $a, b, c, d \\in R$ with $a \\ne c$, there is a unique $x \\in R$ with $T(x, a, b) = T(x, c, d)$;\n4. for all $a, b, c \\in R$, there is a unique $x \\in R$ with $T(a, b, x) = c$;\n5. 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 $T(c, x, y) = d$.\n\n## Remarks\n\nThe intended reading is $T(x, m, b) = x m + b$.\nClause 3 says that two lines of different slopes meet in exactly one point.\nClause 4 says that there is one line of given slope through a given point.\nClause 5 says that two points with different first coordinates lie on exactly one non-vertical line.\n\nExample: a field $k$ with $T(a, b, c) = a b + c$ is a planar ternary ring.\nCheck: clauses 1 and 2 are $a \\cdot 0 + c = 0 \\cdot b + c = c$ and $1 \\cdot a = a \\cdot 1 = a$.\nClause 3 is $x(a - c) = d - b$, solved by $x = (d - b)(a - c)^{-1}$, and clause 4 is solved by $x = c - a b$.\nClause 5 is a $2 \\times 2$ linear system with determinant $a - c \\ne 0$.\n\nNon-example: $\\Z$ with $T(a, b, c) = a b + c$ fails clause 3.\nFor $a = 2$, $b = 0$, $c = 0$, $d = 1$ the equation $2x = 1$ has no solution in $\\Z$.\n\nA planar ternary ring has an addition $a + b = T(a, 1, b)$ and a multiplication $a \\cdot b = T(a, b, 0)$.\nIt is *linear* when $T(a, b, c) = a \\cdot b + c$ for all $a, b, c$.\nChoosing 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.\n",
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