Idempotence
Idempotence (UK: /ˌɪdɛmˈpoʊtəns/,[1] US: /ˌaɪdəm-/)[2] 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[3] 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).
Definition[]
An element x of a set S equipped with a binary operator • is said to be idempotent under • if[4][5]
- x • x = x.
The binary operation • is said to be idempotent if[6][7]
- ∀x∈S, x • x = x.
Examples[]
- In the monoid (ℕ, ×) of the natural numbers with multiplication, only 0 and 1 are idempotent. Indeed, 0 × 0 = 0 and 1 × 1 = 1, which does not hold for other natural numbers.
- In a magma (M, •), an identity element e or an absorbing element a, if it exists, is idempotent. Indeed, e • e = e and a • a = a.
- In a group (G, •), the identity element e is the only idempotent element. Indeed, if x is an element of G such that x • x = x, then x • x = x • e and finally x = e by multiplying on the left by the inverse element of x.
- In the monoids (