by Nate Soares May 9 2016 updated Jun 14 2016

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).


Eric Rogstad

What's the difference between an operator and an operation?

Also what's the difference between an operation and a function?