# Dictionary Definition

idempotent adj : unchanged in value following
multiplication by itself; "this matrix is idempotent"

# User Contributed Dictionary

## English

### Adjective

- Describing an action which, when performed multiple times, has no further effect on its subject after the first time it is performed.

#### Related terms

# Extensive Definition

Idempotence describes the property of operations
in mathematics and
computer
science which yield the same result after the operation is
applied multiple times. The concept of idempotence arises in a
number of places in abstract
algebra (in particular, in the theory of projectors,
closure
operators and functional
programming, in which it is connected to the property of
referential
transparency).

There are several meanings of idempotence,
depending on what the concept is applied to:

- A unary operation (or function) is called idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once. For example, the absolute value function is idempotent as a function from the set of real numbers to the set of real numbers: .
- A binary operation is called idempotent if, whenever it is applied to two equal values, it gives that value as the result. For example, the operation giving the maximum value of two values is idempotent: .
- Given a binary operation, an idempotent element (or simply an idempotent) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result. For example, the number 1 is an idempotent of multiplication: .

## Definitions

### Unary operation

A unary
operation f that is a map from some set S into itself is called
idempotent if, for all x in S,

- f(f(x)) = f(x).

In particular, the identity
function idS, defined by , is idempotent, as is the constant
function Kc, where c is an element of S, defined by .

An important class of idempotent functions is
given by
projections in a vector
space. An example of a projection is the function πxy defined
by , which projects an arbitrary point in 3D space to a point on
the x–y-plane,
where the third coordinate (z) is equal to 0.

### Binary operation

A binary
operation ★ on a set S is called idempotent if, for all x in
S,

- x★x = x.

For example, the operations of set
union and set
intersection are both idempotent, as are logical
conjunction and logical
disjunction, and, in general, the meet
and join
operations of a lattice.

An element x of S is called idempotent for ★ if,
for that element,

- x★x = x.

In particular, an identity
element of ★ is idempotent for the operation.

### Connections

The connections between the three notions are as
follows.

- The statement that the binary operation ★ on a set S is idempotent, is equivalent to the statement that every element of S is idempotent for ★.

- The defining property of unary idempotence, for x in the domain of f, can equivalently be rewritten as , using the binary operation of function composition denoted by "o". Thus, the statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element for the operation o on functions from S to S.

## Common examples

### Computer Science

In computer science, the term idempotent is used to describe method or subroutine calls which can safely be called multiple times, as invoking the procedure a single time or multiple times results in the system maintaining the same state i.e. after the method call all variables have the same value as they did before.Example: Looking up some customer's name and
address are typically idempotent, since the system will not change
state based on this. However, placing an order for a car for the
customer is not, since running the method/call several times will
lead to several orders being placed, and therefore the state of the
system being changed to reflect this.

In Event
Stream Processing, idempotence refers to the ability of a
system to produce the same outcome, even if an event or message is
received more than once.

### Functions

As mentioned above, the identity map and the
constant maps are always idempotent maps. Less trivial examples are
the absolute
value function of a real or
complex
argument, and the floor
function of a real argument.

The function which assigns to every subset U of
some topological
space X the closure
of U is idempotent on the power set of X.
It is an example of a closure
operator; all closure operators are idempotent functions.

### Idempotent ring elements

An idempotent element of a ring
is, by definition, an element which is idempotent with respect to
the ring's multiplication. One may define a partial
order on the idempotents of a ring as follows: if a and b are
idempotents, we write a ≤ b iff ab = ba = a. With respect to
this order, 0 is the smallest and 1 the largest idempotent.

Two idempotents a and b are called orthogonal if
ab = ba = 0. In this case, a + b is also idempotent, and we have a
≤ a + b and b ≤ a + b.

If a is idempotent in the ring R, then so is b =
1 − a; a and b are orthogonal.

If a is idempotent in the ring R, then aRa is
again a ring, with multiplicative identity a.

An idempotent a in R is called central if ax = xa
for all x in R. In this case, Ra is a ring with multiplicative
identity a. The central idempotents of R are closely related to the
decompositions of R as a direct sum of
rings. If R is the direct sum of the rings R1,...,Rn, then the
identity elements of the rings Ri are central idempotents in R,
pairwise orthogonal, and their sum is 1. Conversely, given central
idempotents a1,...,an in R which are pairwise orthogonal and have
sum 1, then R is the direct sum of the rings Ra1,...,Ran. So in
particular, every central idempotent a in R gives rise to a
decomposition of R as a direct sum of Ra and R(1 −
a).

Any idempotent a which is different from 0 and 1
is a zero divisor
(because a(1 − a) = 0). This shows that integral
domains and division
rings don't have such idempotents. Local rings
also don't have such idempotents, but for a different reason. The
only idempotent contained in the Jacobson
radical of a ring is 0. There is a catenoid of idempotents in the
coquaternion
ring.

A ring in which all elements are idempotent is
called a boolean
ring. It can be shown that in every such ring, multiplication
is commutative, and every element is its own additive
inverse.

#### Relation with involutions

If a is an idempotent, then 1-2a is an involution.If b is an involution, then \frac(1-b) is an
idempotent, and these are inverse: thus if 2 is invertible in a
ring, idempotents and involutions are equivalent.

Further, if b is an involution, then \frac(1-b)
and \frac(1+b) are orthogonal idempotents, corresponding to a and
1-a.

#### Numerical examples

One may consider the ring of integers mod n,
where n is squarefree.
By the Chinese
Remainder Theorem, this ring factors into the direct product of
rings of integers mod p. Now each of these factors is a field, so
it's clear that the only idempotents will be 0 and 1. That is, each
factor has 2 idempotents. So if there are m factors, there will be
2^m idempotents.

We can check this for the integers mod 6. Since 6
has 2 factors (2 and 3) it should have 2^2 idempotents.

0 = 0^2 = 0^3 = etc (mod 6) 1 = 1^2 = 1^3 = etc
(mod 6) 3 = 3^2 = 3^3 = etc (mod 6) 4 = 4^2 = 4^3 = etc (mod
6)

### Other examples

Idempotent operations can be found in Boolean
algebra as well.

An idempotent
semiring is a semiring whose addition (not multiplication) is
idempotent.

There are idempotent matrices.

## See also

idempotent in Czech: Idempotence

idempotent in German: Idempotenz

idempotent in Estonian: Idempotentsus

idempotent in Spanish: Idempotente

idempotent in French: Idempotence

idempotent in Italian: Idempotenza

idempotent in Dutch: Idempotentie

idempotent in Japanese: 冪等

idempotent in Polish: Idempotentność

idempotent in Portuguese: Idempotência

idempotent in Russian: Идемпотентный
элемент

idempotent in Serbian: Идемпотенција

idempotent in Finnish: Idempotenssi

idempotent in Swedish: Idempotent

idempotent in Thai: นิจพล

idempotent in Chinese: 等冪