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  "body": "\n## Definition\n\nLet $H$ be a nonempty [[set|0SJ3RX9TXA]] and let $\\mathcal{P}^*(H)$ be the set of nonempty subsets of $H$, a part of the [[power set|00E0YZCERH]].\nA *hyperoperation* on $H$ is a map $\\circ \\colon H \\times H \\to \\mathcal{P}^*(H)$ on the [[Cartesian product|09CEDJYBKP]].\nFor subsets $X, Y \\subseteq H$ write $X \\circ Y = \\bigcup_{x \\in X, y \\in Y} x \\circ y$, and $x \\circ Y = \\set{x} \\circ Y$.\nAn *(algebraic) hyperstructure* is a set $H$ with one or more hyperoperations, possibly together with ordinary operations.\nA set with one hyperoperation $(H, \\circ)$ is a *hypergroupoid*, and it is a *hypergroup* if:\n\n1. $(x \\circ y) \\circ z = x \\circ (y \\circ z)$ for all $x, y, z \\in H$ ([[associativity|0K8F7PVDMX]]);\n2. $x \\circ H = H \\circ x = H$ for all $x \\in H$ (reproduction axiom).\n\nA hypergroupoid that satisfies clause 1 is a *semihypergroup*.\n\n## Remarks\n\nExample: $H = \\R$ with $x \\circ y = [\\min(x, y), \\max(x, y)]$.\nBoth sides of clause 1 equal $[\\min(x, y, z), \\max(x, y, z)]$, since the union of the intervals between $z$ and the points of $[a, b]$ is the interval from $\\min(a, z)$ to $\\max(b, z)$.\nFor clause 2, every $y \\in \\R$ lies in $x \\circ y$, so $x \\circ \\R = \\R \\circ x = \\R$.\n\nNon-example: $H = [0, \\infty)$ with $x \\circ y = [\\max(x, y), \\infty)$.\nClause 1 holds, since both sides equal $[\\max(x, y, z), \\infty)$.\nClause 2 fails: for $x = 1$, $x \\circ H = [1, \\infty) \\ne H$.\nSo this is a semihypergroup but not a hypergroup.\n\nA group $G$ is a hypergroup with $x \\circ y = \\set{xy}$.\n",
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