# Binary operation

### From Online Dictionary of Crystallography

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- | <font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="black">Operazione binaria</font> (''It''). <font color=" | + | <font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="black">Operazione binaria</font> (''It''). <font color="purple">二項演算</font> (''Ja''). <font color="brown">Бинарная операция</font> (''Ru''). <font color="green">Operación binaria</font> (''Sp''). |

## Latest revision as of 17:03, 28 November 2017

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*.