Operator

An operation $~$f$~$ on a 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 closed under $~$f$~$), but that is not always the case.

For 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 $~$+$~$.

The 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$~$).

The number of values that the operator takes as input is called the 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 ring, if we interpret multiplication and subtraction as ring operations).