Binary operation on a set; associativity, commutativity, and a subset closed under the operation

A binary operation on a set $S$ is a function (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain) whose domain is the Cartesian product $S\times S$ (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$),

$\ast :S\times S⟶S,$

written in infix form $x\ast y:=\ast (x,y)$. That the value always lies in $S$ is part of the definition, and is referred to as $S$ being closed under $\ast$. The pair $(S,\ast )$ is called a set equipped with a binary operation.

The operation $\ast$ is

• associative when $(x\ast y)\ast z=x\ast (y\ast z)$ for all $x,y,z\in S$;
• commutative when $x\ast y=y\ast x$ for all $x,y\in S$.

Two elements $x,y\in S$ commute when $x\ast y=y\ast x$; so $\ast$ is commutative exactly when every two elements of $S$ commute.

A subset $T\subseteq S$ is closed under $\ast$ when $x\ast y\in T$ for all $x,y\in T$. In that case the restriction of $\ast$ to $T\times T$ has all its values in $T$ and is therefore a binary operation on $T$, called the restricted operation or the operation induced on $T$; it is again associative if $\ast$ is, and again commutative if $\ast$ is, since both conditions are equations required of elements of $T$, which are in particular elements of $S$.

• Notation. Where no confusion arises the operation is written multiplicatively, $xy$ for $x\ast y$, and additively, $x+y$, when it is commutative and the context is additive. Nothing in this page depends on the choice: every statement is about an abstract operation.

• Bracketing. Associativity is stated for three factors only. That it lets brackets be dropped in a product of any finite length is a theorem, not a restatement: it is Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either, proved below.

• Familiar instances already in the library are addition and multiplication on $\mathbb{N}$ (Addition of natural numbers, Multiplication of natural numbers), on $\mathbb{Z}$ (Arithmetic on the integers) and on any field (Field).