idempotency Idempotency 2002-02-24 16:35:05 2006-07-12 11:52:39 Definition idempotent \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{case} \PMlinkescapeword{cases} \PMlinkescapeword{lattice} \PMlinkescapeword{words} If $(S,*)$ is a magma, then an element $x\in S$ is said to be \emph{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 \PMlinkname{lattice}{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.