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  "body": "\n## Definition\n\nA *shelf* is a [[magma|9689SFPVE9]] $(X, \\triangleright)$ whose operation is [[left self-distributive|VEFE5WZMS3]]:\n$$x \\triangleright (y \\triangleright z) = (x \\triangleright y) \\triangleright (x \\triangleright z) \\quad \\text{for all } x, y, z \\in X.$$\n\n## Remarks\n\nEquivalently, for every $x \\in X$ the left translation $y \\mapsto x \\triangleright y$ is an endomorphism of $(X, \\triangleright)$.\nA shelf is a [[rack|X74VY86TSX]] when every left translation is a bijection.\n\nExample: let $X = \\set{1, 2}$ with $1 \\triangleright y = 2$ and $2 \\triangleright y = y$ for all $y$; this is the Laver table $A_1$.\nIf $x = 2$, both sides equal $y \\triangleright z$.\nIf $x = 1$, the left side is $2$, and the right side is $2 \\triangleright 2 = 2$.\nSo $X$ is a shelf; it is not a rack, since $y \\mapsto 1 \\triangleright y$ is not injective.\n\nNon-example: $(\\Z, +)$, since $x + (y + z) \\ne (x + y) + (x + z)$ for $x \\ne 0$.\n\nSome authors use right self-distributivity instead, with $(z \\triangleleft y) \\triangleleft x = (z \\triangleleft x) \\triangleleft (y \\triangleleft x)$; the two conventions are related by $x \\triangleright y = y \\triangleleft x$.\nIn Mathlib this structure is the class `Shelf`.\n",
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    "name": "Shelf",
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