Binary operation

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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

${\displaystyle S\times S\to S}$ given by ${\displaystyle (x,y)\mapsto x\star y,}$

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

Properties[edit]

A binary operation may satisfy further conditions.

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

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

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