idempotentit

Idempotents, or idempotentit in some languages, describe a property of an operation, element, or function whereby applying it once yields the same result as applying it multiple times. In mathematics, an element e in a set with a binary operation is idempotent if e composed with itself equals e, written e · e = e. This concept appears in structures such as semigroups, monoids, and rings, where idempotents can reveal decompositions or projection-like behaviors.

In linear algebra, a common example is an idempotent matrix P satisfying P^2 = P. Such matrices represent projections onto a subspace along another subspace, and they have eigenvalues in {0,1}. Decompositions arising from idempotents help analyze modules and algebras, including Peirce decompositions in rings.

Beyond pure algebra, idempotence is a functional concept: a function f is idempotent if f(f(x)) = f(x) for all x in its domain. The identity function and any constant function are idempotent; nontrivial idempotent functions occur when the function projects a set onto a subset.

In computer science and distributed systems, idempotence describes operations that can be repeated safely without changing the outcome after the first application. Examples include certain HTTP methods (such as PUT and DELETE in many RESTful designs) and retryable updates or cache operations. Idempotence improves robustness in the face of retries and partial failures, distinguishing it from invertibility, where an operation has a reverse, not always possible for repeated applications.

Category-theoretically, an endomorphism p with p ∘ p = p is idempotent, and such morphisms may split, yielding further structural insights.