idempotenter

Idempotenter, or idempotent elements, are elements e in a set equipped with a binary operation (such as a semigroup, ring, or algebra) that satisfy e^2 = e. Equivalently, applying the operation to e with itself yields e again. In category theory an idempotent endomorphism f: X → X satisfies f ∘ f = f.

In algebra, idempotent elements act as projections. The most familiar examples are 0 and 1 in any ring. A matrix P is idempotent if P^2 = P; such matrices are called projection matrices. They represent linear projections onto the image of P. Over a field, every idempotent matrix is diagonalizable with eigenvalues 0 or 1.

In functional terms, a function f: X → X is idempotent if f(f(x)) = f(x) for all x. Idempotent maps occur as projection operators in linear spaces or as simplifications of state transitions. In computer science and web services, an operation is idempotent if repeating it produces the same effect as a single application (for example, HTTP GET or PUT requests, or certain database migrations).

Additional context: In functional analysis and operator theory, idempotent operators are projections onto a subspace; in category theory the notion leads to the idea of splitting idempotents, a construction related to decomposing objects into a part preserved by the idempotent and a complementary part.

Idempotenter are widely studied in algebra, topology, and computer science due to their stable, projection-like behavior.