idempotency
If (S,*) is a magma, then an element x∈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 ∧ and ∨ operations
in a lattice
(http://planetmath.org/Lattice) are idempotent, because x∧x=x and x∨x=x for all x in the lattice.
A function f:D→D is said to be idempotent if f∘f=f. (This is just a special case of the first definition above, the magma in question being (DD,∘), 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∈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 |