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 $\wedge$ and $\vee$ operations in a lattice (http://planetmath.org/Lattice) are idempotent, because $x\wedge x=x$ and $x\vee x=x$ for all $x$ in the lattice.

A function $f:D\to D$ 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
Canonical name	Idempotency
Date of creation	2013-03-22 12:27:31
Last modified on	2013-03-22 12:27:31
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	21
Author	yark (2760)
Entry type	Definition
Classification	msc 20N02
Related topic	BooleanRing
Related topic	PeriodOfMapping
Related topic	Idempotent2
Defines	idempotent