# Operator

https://arbital.com/p/operator_mathematics

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).