# Binary operation

### From Online Dictionary of Crystallography

Loi de composition (*Fr*). Zweistellige Verknüpfung (*Ge*). Operazione binaria (*It*). 二項演算 (*Ja*). Бинарная операция (*Ru*). Operación binaria (*Sp*).

A **binary operation** on a set *S* is a mapping *f* from the Cartesian product *S × S* to *S*. A mapping from *K x S* to *S*, where *K* need not be *S*, is called an **external binary operation**.

Many binary operations are commutative [*i.e.* *f(a,b) = f(b,a)* holds for all *a, b* in *S*] or associative [*i.e.* *f(f(a,b), c) = f(a, f(b,c))* holds for all *a,b,c* in *S*]. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations.

Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@) and composition.

Binary operations are often written using infix notation such as *a * b*, *a + b* or *a · b*, rather than by functional notation of the form *f(a,b)*. Sometimes they are even written just by concatenation: *ab*.