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  "body": "\n## Definition\n\nLet $X$ be a set.\nThe *associativity equation* on $X$ is the [[functional equation|SRAN4EKWS0]]\n$$F(F(x, y), z) = F(x, F(y, z)) \\quad \\text{for all } x, y, z \\in X,$$\nfor an unknown [[binary operation|0M418FE7BB]] $F \\colon X \\times X \\to X$.\nA solution is an associative operation on $X$.\n\n## Remarks\n\nExample: $F(x, y) = x + y + xy$ on $X = (-1, \\infty)$ is a solution, since both sides equal $(1 + x)(1 + y)(1 + z) - 1$.\nNon-example: $F(x, y) = x - y$ on $\\R$ is not a solution, since $F(F(0, 0), 1) = -1$ while $F(0, F(0, 1)) = 1$.\nEvery operation $F(x, y) = g^{-1}(g(x) + g(y))$, with $g \\colon X \\to \\R$ injective and $g(X)$ closed under addition, is a solution.\nOn a real interval, the continuous solutions that are strictly increasing in each variable are all of this form, with $g$ continuous and strictly monotone.\n",
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