# Binary operation

### From Online Dictionary of Crystallography

##### Revision as of 09:16, 29 May 2007 by MassimoNespolo (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 function *f* from the Cartesian product S × S to S. A binary function from K and S to S, where K need not be S, is called an ** external binary operation **.

Many binary operations are commutative or associative. 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 on a single set.

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

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 juxtaposition: ab.