Idempotent element

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In algebra, an idempotent element with respect to a binary operation is an element which is unchanged when combined with itself.

Formally, let be a binary operation on a set X. An element E of X is an idempotent for if

Examples include an identity element or an absorbing element. An important class of examples is formed by considering operators on a set (functions from a set to itself) under function composition: for example, endomorphisms of a vector space. Here the idempotents are projections, corresponding to direct sum decompositions. For example, the idempotent matrix

is an idempotent for matrix multiplication corresponding to the operation of projection onto the x-axis along the y-axis.

An idempotent binary operation is one for which every element is an idempotent.