Difference between revisions of "Binary operation"

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(Properties: Added idempotent)
 
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* [[Power-associative]]: <math>(x \star x) \star x = x \star (x \star x)</math>
 
* [[Power-associative]]: <math>(x \star x) \star x = x \star (x \star x)</math>
  
Special elements which may be associated with a binary operations include:
+
Special elements which may be associated with a binary operation include:
 
* [[Neutral element]] ''I'': <math>I \star x = x \star I = x</math> for all ''x''
 
* [[Neutral element]] ''I'': <math>I \star x = x \star I = x</math> for all ''x''
 
* [[Absorbing element]] ''O'': <math>O \star x = x \star O = O</math> for all ''x''
 
* [[Absorbing element]] ''O'': <math>O \star x = x \star O = O</math> for all ''x''
 +
* [[Idempotent element]] ''E'': <math>E \star E = E</math>

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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 on a set S is a function on the Cartesian product

given by

using operator notation rather than functional notation, which would call for writing .

Properties

A binary operation may satisfy further conditions.

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