Binary operation

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In mathematics, a binary operation on a set is a function of two variables which assigns a value to any pair of elements of the set: principal motivating examples include the arithmetic and elementary algebraic operations of addition, subtraction, multiplication and division.

Formally, a binary operation  ⋆ {\displaystyle \star } on a set S is a function on the Cartesian product

S × S -> S {\displaystyle S\times S\rightarrow S\,} given by  ( x , y ) ↦ x ⋆ y , {\displaystyle (x,y)\mapsto x\star y,\,}

using operator notation rather than functional notation, which would call for writing  ⋆ ( x , y ) {\displaystyle \star (x,y)} .

## Properties[edit]

A binary operation may satisfy further conditions.

* Commutative:  x ⋆ y = y ⋆ x {\displaystyle x\star y=y\star x}
* Associative:  ( x ⋆ y ) ⋆ z = x ⋆ ( y ⋆ z ) {\displaystyle (x\star y)\star z=x\star (y\star z)}
* Alternative:  ( x ⋆ y ) ⋆ y = x ⋆ ( y ⋆ y ) {\displaystyle (x\star y)\star y=x\star (y\star y)}
* Power-associative:  ( x ⋆ x ) ⋆ x = x ⋆ ( x ⋆ x ) {\displaystyle (x\star x)\star x=x\star (x\star x)}

Special elements which may be associated with a binary operation include:

* Neutral element I:  I ⋆ x = x ⋆ I = x {\displaystyle I\star x=x\star I=x} for all x
* Absorbing element O:  O ⋆ x = x ⋆ O = O {\displaystyle O\star x=x\star O=O} for all x
* Idempotent element E:  E ⋆ E = E {\displaystyle E\star E=E}