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text: 'An operation $f$ on a [3jz set] $S$ is a function that takes some values from $S$ and produces a new value. An operation can take any number of values from $S$, including zero (in which case $f$ is simply a constant) or infinitely many (in which case we call $f$ an "infinitary operation"). Common operations take a finite non-zero number of parameters. Operations often produce a value that is also in $S$ (in which case we say $S$ is [3gy closed] under $f$), but that is not always the case.\n\nFor example, the function $+$ is a binary operation on [45h $\\mathbb N$], meaning it takes two values from $\\mathbb N$ and produces another. Because $+$ produces a value that is also in $\\mathbb N$, we say that $\\mathbb N$ is closed under $+$.\n\nThe function $\\operatorname{neg}$ that maps $x$ to $-x$ is a unary operation on [48l $\\mathbb Z$]: It takes one value from $\\mathbb Z$ as input, and produces an output in $\\mathbb Z$ (namely, the negation of the input). $\\operatorname{neg}$ is also a unary operation on $\\mathbb N$, but $\\mathbb N$ is not closed under $\\operatorname{neg}$ (because $\\operatorname{neg}(3)=-3$ is not in $\\mathbb N$).\n\nThe number of values that the operator takes as input is called the [3h8 arity] of the operator. For example, the function $\\operatorname{zero}$ which takes no inputs and returns $0$ is a zero-arity operator; and the operator $f(a, b, c, d) = ac - bd$ is a four-arity operator (which can be used on any [3gq ring], if we interpret multiplication and subtraction as ring operations).',
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