# Binary operation

### From Online Dictionary of Crystallography

##### Revision as of 14:02, 1 April 2009 by BerndSouvignier (talk | contribs)

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

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*.