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 $A \times B := \{\, z \in \mathcal{P}(\mathcal{P}(A \cup B)) : \exists a \in A\ \exists b \in B\ z = (a,b) \,\}$),

$* \;:\; S \times S \longrightarrow S,$

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

The operation $*$ is

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

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

A subset $T \subseteq S$ is closed under $*$ when $x * y \in T$ for all $x, y \in T$. In that case the restriction of $*$ 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 $*$ is, and again commutative if $*$ 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 $x * y$, 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).