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 $N$, meaning it takes two values from $N$ and produces another. Because $+$ produces a value that is also in $N$, we say that $N$ is closed under $+$.

The function $\mathrm{neg}$ that maps $x$ to $-x$ is a unary operation on $Z$: It takes one value from $Z$ as input, and produces an output in $Z$ (namely, the negation of the input). $\mathrm{neg}$ is also a unary operation on $N$, but $N$ is not closed under $\mathrm{neg}$ (because $\mathrm{neg}\left(3\right)=-3$ is not in $N$).

The number of values that the operator takes as input is called the arity of the operator. For example, the function $\mathrm{zero}$ 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).