View source for Idempotence

'''Idempotence''' ({{IPAc-en|ˌ|aɪ|d|ɨ|m|ˈ|p|oʊ|t|ən|s}} {{respell|EYE|dəm|POH|təns}}{{fact|date=July 2015}}) is the property of certain [[operation (mathematics)|operations]] in [[mathematics]] and [[computer science]], that can be applied multiple times without changing the result beyond the initial application.

(TO DO: fix template.)

== Description ==

The concept of idempotence arises in a number of places in [[abstract algebra]] (in particular, in the theory of [[projector (linear algebra)|projector]]s and [[closure operator]]s) and [[functional programming]] (in which it is connected to the property of [[Referential transparency (computer science)|referential transparency]]).

The term was introduced by [[Benjamin Peirce]]<ref>Polcino & Sehgal (2002), p. 127.</ref> in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from ''[[wikt:idem|idem]]'' + ''[[wikt:potence|potence]]'' (same + power).

There are several meanings of idempotence, depending on what the concept is applied to:

*A [[unary operation]] (or [[function (mathematics)|function]]) is idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once; i.e., {{nowrap|''ƒ''(''ƒ''(''x'')) ≡ ''ƒ''(''x'')}}. For example, the [[absolute value]] function, where {{nowrap|abs(abs(''x'')) ≡ abs(''x'')}}.
*A [[binary operation]] is idempotent if, whenever it is applied to two equal values, it gives that value as the result. For example, the function giving the [[maximum]] value of two equal values is idempotent: {{nowrap|max (''x'', ''x'') ≡ ''x''}}.

*Given a binary operation, an idempotent element (or simply an "idempotent") for the operation is a value for which the operation, when given that value for both of its operands, gives that value as the result. For example, the number 1 is an idempotent of [[multiplication]]: {{nowrap|1 × 1 {{=}} 1}}.

== See also ==

* [[Mathematics]]
* [[Computer science]]
* [[Identification scheme]]
* [[Unique identifier]]

== External links == 

* [https://en.wikipedia.org/wiki/Idempotence Idempotence] @ Wikipedia