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  "body": "\n## Example\n\nLet $S$ be a set with a Steiner triple system on it, that is, a [[Steiner system|7BPSC4T4M3]] $S(2, 3, n)$.\nIts blocks are $3$-element subsets of $S$, and every two distinct points lie in exactly one block.\nThe *Steiner quasigroup* of this system is $S$ with the binary operation $\\cdot$ given by:\n\n1. $x \\cdot x = x$ for every $x \\in S$;\n2. $x \\cdot y = z$ for $x \\ne y$, where $\\set{x, y, z}$ is the unique block that contains $x$ and $y$.\n\nThis operation is [[commutative|XAC78GN2YJ]] and [[idempotent|9FS1JFZSX4]], and it satisfies $x \\cdot (x \\cdot y) = y$ for all $x, y$.\nFor $x \\ne y$ the last law holds because $x \\cdot y = z$ and $x \\cdot z = y$ come from the same block.\n\nCheck that $S$ is a quasigroup, that is, every equation $a \\cdot y = b$ and $y \\cdot a = b$ has exactly one solution $y$.\nThe law $x \\cdot (x \\cdot y) = y$ says that the map $y \\mapsto a \\cdot y$ is its own inverse, so it is a bijection.\nHence the equation $a \\cdot y = b$ has the unique solution $y = a \\cdot b$.\nBy commutativity the equation $y \\cdot a = b$ has the same unique solution.\n\n## Remarks\n\nThe smallest case is $S = \\set{0, 1, 2}$ with the single block $\\set{0, 1, 2}$.\nThere the operation is $x \\cdot y = -x - y \\bmod 3$.\nFor instance $1 \\cdot 2 = -3 \\equiv 0$, which is the third point of the block, and $1 \\cdot 1 = -2 \\equiv 1$.\nConversely, a commutative idempotent quasigroup that satisfies $x \\cdot (x \\cdot y) = y$ comes from a Steiner triple system, whose blocks are the sets $\\set{x, y, x \\cdot y}$ with $x \\ne y$.\nThe multiplication table of a Steiner quasigroup is a symmetric [[Latin square|CDVRMVED2V]] with entry $x$ in row $x$ and column $x$.\n\n## Explanations\n\n### L1 human\n\nA Steiner triple system groups points into triangles so that any two points share exactly one triangle.\nThe Steiner quasigroup turns this into a multiplication: the product of two points is the third corner of their triangle.\nThe product of a point with itself is the point.\nMultiplying twice by the same point brings you back, which makes every equation $a \\cdot y = b$ solvable in exactly one way.\nOn the three points $0, 1, 2$ the product of two different points is the remaining one.\n",
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