Idempotence: Difference between revisions
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In [[mathematics]] '''idempotence''' is the property of an [[operation (mathematics)|operation]] that repeated application has no effect. | In [[mathematics]] '''idempotence''' is the property of an [[operation (mathematics)|operation]] that repeated application has no further effect. | ||
A [[binary operation]] <math>\star</math> is ''idempotent'' if | A [[binary operation]] <math>\star</math> is ''idempotent'' if | ||
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equivalently, every element is an [[idempotent element]] for <math>\star</math>. | equivalently, every element is an [[idempotent element]] for <math>\star</math>. | ||
Examples of idempotent binary operations include [[join]] and [[meet]] in a [[lattice (order)|lattice]]. | Examples of idempotent binary operations include [[join]] and [[meet]] in a [[lattice (order)|lattice]]; [[union]] and [[intersection]] on [[set (mathematics)|sets]]; [[disjunction]] and [[conjunction]] in [[propositional logic]]. | ||
A [[unary operation]] (function from a set to itself) π is idempotent if it is an idempotent element for [[function composition]], π<sup>2</sup> = π. | A [[unary operation]] (function from a set to itself) π is idempotent if it is an idempotent element for [[function composition]], π<sup>2</sup> = π. | ||
Revision as of 10:50, 23 December 2008
In mathematics idempotence is the property of an operation that repeated application has no further effect.
A binary operation is idempotent if
- for all x:
equivalently, every element is an idempotent element for .
Examples of idempotent binary operations include join and meet in a lattice; union and intersection on sets; disjunction and conjunction in propositional logic.
A unary operation (function from a set to itself) π is idempotent if it is an idempotent element for function composition, π2 = π.