# 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 (http://planetmath.org/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.

Title idempotency Idempotency 2013-03-22 12:27:31 2013-03-22 12:27:31 yark (2760) yark (2760) 21 yark (2760) Definition msc 20N02 BooleanRing PeriodOfMapping Idempotent2 idempotent