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  "body": "\n## Definition\n\nLet $A$ be a set and let $I$ be an infinite set.\nWrite $A^I$ for the [[Cartesian product|09CEDJYBKP]] of copies of $A$ indexed by $I$, the set of families $(a_i)_{i \\in I}$ of elements of $A$.\nAn *infinitary operation* on $A$ with index set $I$ is a [[function|0EPAT51DXQ]] $f \\colon A^I \\to A$.\nIts [[arity|MSXX2P86CY]] is the cardinal $\\abs{I}$.\n\n## Remarks\n\nExample: on $[0, 1]$, the map $s \\colon [0,1]^{\\N} \\to [0, 1]$, $s((x_n)_{n \\in \\N}) = \\sup_n x_n$, is an infinitary operation of arity $\\aleph_0$.\nEach sequence in $[0, 1]$ is bounded above by $1$, so its supremum exists and lies in $[0, 1]$.\n\nNon-example: the map $(x_n)_{n \\in \\N} \\mapsto \\sum_{n} x_n$ is not an infinitary operation on $\\R$, because it is not defined on the whole of $\\R^{\\N}$: the series $\\sum_n 1$ diverges.\n\nSome authors index the arguments by an infinite ordinal $\\alpha$ instead of an arbitrary set $I$, and call $\\alpha$ the arity.\n",
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