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 \star be a binary operation on a set X. An element E of X is an idempotent for \star if

E \star E = E . \,

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

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}

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.

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