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  "body": "\n## Summary\n\nA quasifield is a set with an addition and a multiplication that behaves like a [[division ring|0EA5BTA0CX]].\nThe difference is that multiplication need not be associative and distributes only on one side.\nQuasifields are the coordinate systems of [[translation planes|JZ5J6REAMD]]: every quasifield gives a translation plane and every translation plane comes from a quasifield.\n[[Semifields|AHH97EJ6VH]], finite [[near-fields|DSE72WQ5A2]] and division rings are special quasifields.\n\n## Explanations\n\n### L1 human\n\nIn a field one can add, subtract, multiply, divide, and the usual rules hold.\nA quasifield keeps the addition and the possibility to divide, but allows the multiplication to break associativity and one of the two distributive laws.\nSuch loose arithmetic is still enough to draw a plane in which lines have equations like $y = x \\cdot m + k$.\nThe Hall plane of order $9$ comes from a quasifield with $9$ elements that is not a field.\n",
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