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  "body": "\n## Summary\n\nThe associativity equation is the [[functional equation|SRAN4EKWS0]] $F(F(x, y), z) = F(x, F(y, z))$ for an unknown binary operation $F$ on a set.\nIts solutions are exactly the associative [[binary operations|0M418FE7BB]], so it turns the algebraic law of [[associativity|0K8F7PVDMX]] into a problem about unknown functions.\nOn a real interval, its continuous solutions that are strictly increasing in each variable are the operations $F(x, y) = g^{-1}(g(x) + g(y))$ with $g$ continuous and strictly monotone.\nSo, up to a change of scale, every such operation on an interval is ordinary addition.\n\n## Explanations\n\n### L1 human\n\nAssociativity says that brackets do not matter when one combines three things.\nThe equation asks which formulas $F(x, y)$ have this property.\nAddition $x + y$ is the basic solution, and every regular solution on an interval is addition in disguise, after a change of scale $g$.\nFor example, $F(x, y) = x + y + xy$ on $(-1, \\infty)$ is associative, because $1 + F(x, y) = (1 + x)(1 + y)$.\n",
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