# Difference between revisions of "Binary operation"

### From Online Dictionary of Crystallography

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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'''. | 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 | + | 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(@) | + | 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'' | + | 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''. |

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Revision as of 12:48, 12 May 2017

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*.