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'idempotency'
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| Title of object: |
idempotency |
| Canonical Name: |
Idempotency |
| Type: |
Definition |
| Created on: |
2002-02-24 16:35:05 |
| Modified on: |
2003-12-05 04:52:46 |
| Classification: |
msc:26A30 |
| Synonyms: |
idempotency=idempotent |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts} |
Content:
\PMlinkescapeword{lattice}
If $(S,*)$ is a magma, then an element $x\in S$ is said to be \emph{idempotent} if $x*x=x$. If every element of $S$ 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 above definition, 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 iff 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. |
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