If (S,*) is a magma, then an element xS is said to be idempotentMathworldPlanetmathPlanetmath if x*x=x. For example, every identity elementMathworldPlanetmath is idempotent, and in a group this is the only idempotent element. An idempotent element is often just called an idempotent.

If every element of the magma (S,*) is idempotent, then the binary operationMathworldPlanetmath * (or the magma itself) is said to be idempotent. For example, the and operationsMathworldPlanetmath in a latticeMathworldPlanetmath ( are idempotent, because xx=x and xx=x for all x in the lattice.

A function f:DD is said to be idempotent if ff=f. (This is just a special case of the first definition above, the magma in question being (DD,), the monoid of all functions from D to D with the operation of function composition.) In other words, f is idempotent if and only if repeated application of f has the same effect as a single application: f(f(x))=f(x) for all xD. An idempotent linear transformation from a vector spaceMathworldPlanetmath to itself is called a projectionPlanetmathPlanetmath.

Title idempotency
Canonical name Idempotency
Date of creation 2013-03-22 12:27:31
Last modified on 2013-03-22 12:27:31
Owner yark (2760)
Last modified by yark (2760)
Numerical id 21
Author yark (2760)
Entry type Definition
Classification msc 20N02
Related topic BooleanRing
Related topic PeriodOfMapping
Related topic Idempotent2
Defines idempotent