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 ⋆ on a set S is a function on the Cartesian product

S×S->S given by (x,y)↦x⋆y,

using operator notation rather than functional notation, which would call for writing ⋆(x,y).

## Properties[edit]

A binary operation may satisfy further conditions.

* Commutative: x⋆y=y⋆x
* Associative: (x⋆y)⋆z=x⋆(y⋆z)
* Alternative: (x⋆y)⋆y=x⋆(y⋆y)
* Power-associative: (x⋆x)⋆x=x⋆(x⋆x)

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

* Neutral element I: I⋆x=x⋆I=x for all x
* Absorbing element O: O⋆x=x⋆O=O for all x
* Idempotent element E: E⋆E=E