Idempotent element

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.