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  "body": "\n## Summary\n\nA 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.\nMarshall Hall showed that every [[projective plane|DANFNN8C4Y]] can be coordinatized by a planar ternary ring.\nConversely, every planar ternary ring gives a projective plane.\nAlgebraic properties of the ternary ring match geometric properties of the plane.\nFor instance, the plane is a [[Desarguesian plane|HPX85PA8NX]] when the ring is a division ring with $T(x, m, b) = x m + b$.\nIt is the main tool for constructing and classifying [[non-Desarguesian planes|HYE5FT4M34]].\n\n## Explanations\n\n### L1 human\n\nIn the usual plane, a non-vertical line is the set of points with $y = x m + b$.\nAn abstract projective plane may have no multiplication and no addition.\nStill, a rule tells which $y$ belongs to $x$ on the line with slope $m$ and intercept $b$.\nThat rule is the ternary operation $T(x, m, b)$.\nIts axioms say exactly what is needed for two points to determine one line and two lines to meet once.\n",
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