# 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 |