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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]] and let $*$ be a [[binary operation|0M418FE7BB]] on $X$.\nThe operation $*$ is *left self-distributive* if\n$$x * (y * z) = (x * y) * (x * z) \\quad \\text{for all } x, y, z \\in X.$$\nIt is *right self-distributive* if\n$$(y * z) * x = (y * x) * (z * x) \\quad \\text{for all } x, y, z \\in X.$$\nIt is *self-distributive* if it is left or right self-distributive, as fixed by the context, and in other texts if it is both.\n\n## Remarks\n\nLeft self-distributivity says that $*$ has the [[distributive property|63JGSKKF7J]] over itself from the left.\nEquivalently, each left translation $L_x \\colon y \\mapsto x * y$ is an endomorphism of $(X, *)$.\n\nExample: on $\\Z$ let $x * y = 2x - y$.\nThen $x * (y * z) = 2x - (2y - z) = 2x - 2y + z$ and $(x * y) * (x * z) = 2(2x - y) - (2x - z) = 2x - 2y + z$, so $*$ is left self-distributive.\n\nExample: in a group, $x * y = x y x^{-1}$ is left self-distributive, since both sides equal $x y z y^{-1} x^{-1}$.\n\nNon-example: addition on $\\Z$.\nHere $x + (y + z) = x + y + z$ while $(x + y) + (x + z) = 2x + y + z$, which differ when $x \\ne 0$.\n",
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