Idempotent

Idempotent describes an operation that yields the same result when applied multiple times as when applied once. In mathematics, a function f is idempotent if f(f(x)) = f(x) for all x in its domain. In algebra, an element e of a semigroup is idempotent if e * e = e, where * denotes the operation of the semigroup.

In linear algebra, an idempotent linear map P satisfies P^2 = P. Such operators are projections onto a subspace, and their eigenvalues are restricted to 0 or 1. A matrix with A^2 = A is called an idempotent matrix; it represents a projection in the ambient space, and its rank equals its trace.

In category theory, an endomorphism e: X → X is idempotent if e ∘ e = e. Idempotents can be studied via the construction of idempotent completion (Karoubi envelope), which formally splits idempotent morphisms.

In computer science, idempotence refers to the property that repeated application of an operation has no additional effect after the first. This is valuable for safe retries in distributed systems and error handling. In the HTTP protocol, methods such as GET, PUT, DELETE, and OPTIONS are defined to be idempotent, while POST is typically not. Database operations can be designed to be idempotent (for example, upsert operations) to avoid duplicate effects from retries.

Common examples of idempotent operations include taking the minimum or maximum of a set of values, or applying a function like x → max(x, c) where the result stabilizes after one application. Idempotence provides a mathematical and practical lens for reasoning about stability and repeatability across disciplines.