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  "body": "\n## Definition\n\nLet $\\Sigma$ be a signature, a set of function symbols each with an [[arity|MSXX2P86CY]] $n \\ge 0$.\nFor a [[set|0SJ3RX9TXA]] $A$, let $\\mathcal{P}^*(A)$ be the set of nonempty subsets of $A$, a part of the [[power set|00E0YZCERH]].\nA *multialgebra* of signature $\\Sigma$ is a set $A$ together with a map $f^A \\colon A^n \\to \\mathcal{P}^*(A)$ for each symbol $f \\in \\Sigma$ of arity $n$.\nHere $A^n$ is the [[Cartesian power|09CEDJYBKP]] of $A$.\nEach map $f^A$ is a *multioperation* on $A$.\n\n## Remarks\n\nAn ordinary algebra is the same as a multialgebra in which every value $f^A(a_1, \\dots, a_n)$ has exactly one element.\nSome authors allow the empty set as a value; then partial algebras are also multialgebras.\n\nExample: $A = \\R$ with the binary multioperation $x \\circ y = [\\min(x, y), \\max(x, y)]$.\nEach value is a closed interval that contains $x$, so it is nonempty.\n\nNon-example: $A = \\set{0, 1, 2}$ with $x \\circ y = \\set{z \\in A : z \u003e x + y}$.\nFor $x = y = 1$ the value is $\\set{z \\in A : z \u003e 2} = \\emptyset$, so $\\circ$ is not a multioperation in the sense above.\n",
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