Idempotence

Idempotence (UK: /ˌɪdɛmˈpoʊtəns/, US: /ˌaɪdəm-/) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). Idempotence (UK: /ˌɪdɛmˈpoʊtəns/, US: /ˌaɪdəm-/) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by Benjamin Peirce 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 idem + potence (same + power). An element x of a magma (M, •) is said to be idempotent if: If all elements are idempotent with respect to •, then • is called idempotent.The formula ∀x, x • x = x is called the idempotency law for •. In the monoid (FE, ∘) of the functions from a set E to a subset F of E with the function composition ∘, idempotent elements are the functions f: E → F such that f ∘ f = f, in other words such that for all x in E, f(f(x)) = f(x) (the image of each element in E is a fixed point of f). For example: