idempotency

If $(S,*)$ is a magma, then an element $x\in S$ is said to be idempotent if $x*x=x$ . For example, every identity element 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 operation $*$ (or the magma itself) is said to be idempotent. For example, the $\land$ and $\lor$ operations in a lattice are idempotent, because $x\land x = x$ and $x\lor x = x$ for all $x$ in the lattice.

A function $f\colon D\to D$ is said to be idempotent if $f\circ f=f$ . (This is just a special case of the first definition above, the magma in question being $(D^D,\circ)$ , 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 $x\in D$ . An idempotent linear transformation from a vector space to itself is called a projection.