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 idemptotent 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.