Binary operation

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 $\star$ on a set S is a function on the Cartesian product

$S\times S\rightarrow S\,$ given by $(x,y)\mapsto x\star y,\,$

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

Properties

A binary operation may satisfy further conditions.

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

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

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