Editing Idempotence (section)

== Examples ==
* In the [[monoid]] <math>(\mathbb{N}, \times)</math> of the [[natural number]]s with [[multiplication]], only <math>0</math> and <math>1</math> are idempotent. Indeed, {{nowrap|1=<math>0\times 0 = 0</math>}} and {{nowrap|1=<math>1\times 1=1</math>}}.
* In the [[monoid]] <math>(\mathbb{N}, +)</math> of the [[natural number]]s with [[addition]], only <math>0</math> is idempotent. Indeed, {{nowrap|1=0 + 0 = 0}}.
* In a [[Magma (algebra)|magma]] <math>(M, \cdot)</math>, an [[identity element]] <math>e</math> or an [[absorbing element]] <math>a</math>, if it exists, is idempotent. Indeed, {{nowrap|1=<math>e\cdot e=e</math>}} and {{nowrap|1=<math>a\cdot a=a</math>}}.
* In a [[Group (mathematics)|group]] <math>(G, \cdot)</math>, the identity element <math>e</math> is the only idempotent element. Indeed, if <math>x</math> is an element of <math>G</math> such that {{nowrap|1=<math>x\cdot x=x</math>}}, then {{nowrap|1=<math>x\cdot x=x\cdot e</math>}} and finally <math>x=e</math> by multiplying on the left by the [[inverse element]] of <math>x</math>.
* In the monoids <math>(\mathcal{P}(E), \cup)</math> and <math>(\mathcal{P}(E), \cap)</math> of the [[power set]] <math>\mathcal{P}(E)</math> of the set <math>E</math> with [[Union (set theory)|set union]] <math>\cup</math> and [[Intersection (set theory)|set intersection]] <math>\cap</math> respectively, <math>\cup</math> and <math>\cap</math> are idempotent. Indeed, {{nowrap|1=<math>x\cup x=x</math> for all <math>x\in \mathcal{P}(E)</math>}}, and {{nowrap|1=<math>x\cap x=x</math> for all <math>x\in \mathcal{P}(E)</math>}}.
* In the monoids <math>(\{0, 1\}, \vee)</math> and <math>(\{0, 1\}, \wedge)</math> of the [[Boolean domain]] with [[logical disjunction]] <math>\vee</math> and [[logical conjunction]] <math>\wedge</math> respectively, <math>\vee</math> and <math>\wedge</math> are idempotent. Indeed, {{nowrap|1=<math>x\vee x=x</math> for all <math>x\in \{0, 1\}</math>}}, and {{nowrap|1=<math>x\wedge x=x</math> for all <math>x\in \{0, 1\}</math>}}.
* In a [[GCD domain]] (for instance in <math>\mathbb{Z}</math>), the operations of [[greatest common divisor|GCD]] and [[least common multiple|LCM]] are idempotent.
* In a [[Boolean ring]], multiplication is idempotent.
* In a [[Tropical semiring]], addition is idempotent.
* In a [[Matrix_ring|ring of quadratic matrices]], the [[determinant]] of an [[idempotent matrix]] is either 0 or 1. If the determinant is 1, the matrix necessarily is the [[identity matrix]].{{cn|date=November 2022}}

===Idempotent functions===
In the monoid <math>(E^E, \circ)</math> of the functions from a set <math>E</math> to itself (see [[Function_(mathematics)#Set_exponentiation|set exponentiation]]) with [[function composition]] <math>\circ</math>, idempotent elements are the functions {{nowrap|<math>f\colon E\to E</math>}} such that {{nowrap|1=<math>f\circ f = f</math>}},{{efn |This is an equation between functions. Two functions are equal if their domains and ranges agree, and their output values agree on their whole domain.}} that is such that {{nowrap|1=<math>f(f(x))=f(x)</math> for all <math>x\in E</math>}} (in other words, the image <math>f(x)</math> of each element <math>x\in E</math> is a [[Fixed point (mathematics)|fixed point]] of <math>f</math>). For example:
* the [[absolute value]] is idempotent. Indeed, {{nowrap|1=<math>\operatorname{abs}\circ \operatorname{abs}=\operatorname{abs}</math>}}, that is {{nowrap|1=<math>\operatorname{abs}(\operatorname{abs}(x))=\operatorname{abs}(x)</math> for all <math>x</math>}};
* [[Constant function|constant]] functions are idempotent;
* the [[identity function]] is idempotent;
* the [[Floor and ceiling functions|floor]], [[Floor and ceiling functions|ceiling]] and [[fractional part]] functions are idempotent;
* the real part function <math>\mathrm{Re}(z)</math> of a [[complex number]], is idempotent.
* the [[Generating set of a group|subgroup generated]] function from the power set of a group to itself is idempotent;
* the [[convex hull]] function from the power set of an [[affine space]] over the [[Real number|reals]] to itself is idempotent;
* the [[Closure (topology)|closure]] and [[Interior (topology)|interior]] functions of the power set of a [[topological space]] to itself are idempotent;
* the [[Kleene star]] and [[Kleene plus]] functions of the power set of a monoid to itself are idempotent;
* the idempotent [[endomorphism]]s of a [[vector space]] are its [[Projection (linear algebra)|projections]].

If the set <math>E</math> has <math>n</math> elements, we can partition it into <math>k</math> chosen fixed points and {{nowrap|<math>n-k</math>}} non-fixed points under <math>f</math>, and then <math>k^{n-k}</math> is the number of different idempotent functions. Hence, taking into account all possible partitions,
: <math>\sum_{k=0}^n {n \choose k} k^{n-k}</math>
is the total number of possible idempotent functions on the set. The [[integer sequence]] of the number of idempotent functions as given by the sum above for ''n'' = 0, 1, 2, 3, 4, 5, 6, 7, 8, ... starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, ... {{OEIS|A000248}}.

Neither the property of being idempotent nor that of being not is preserved under function composition.{{efn |If <math>f</math> and <math>g</math> commute under composition (i.e. if {{nowrap|1=<math>f\circ g=g\circ f</math>}}) then idempotency of both <math>f</math> and <math>g</math> implies that of {{nowrap|<math>f\circ g</math>}}, since {{nowrap|1=<math>(f\circ g)\circ (f\circ g)=f\circ (g\circ f)\circ g=f\circ (f\circ g)\circ g=(f\circ f)\circ (g\circ g)=f\circ g</math>}}, using the associativity of composition.}} As an example for the former, {{nowrap|1=<math>f(x)=x</math>}} [[modular arithmetic|mod]] 3 and <math>g(x)=\max(x, 5)</math> are both idempotent, but {{nowrap|<math>f\circ g</math>}} is not,{{efn |e.g. <math>f(g(7))=f(7)=1</math>, but <math>f(g(1))=f(5)=2\neq 1</math>}} although {{nowrap|<math>g\circ f</math>}} happens to be.{{efn |Also showing that commutation of <math>f</math> and <math>g</math> is not a [[necessary condition]] for idempotency preservation.}} As an example for the latter, the negation function <math>\neg</math> on the Boolean domain is not idempotent, but {{nowrap|<math>\neg\circ\neg</math>}} is. Similarly, unary negation {{nowrap|<math>-(\cdot)</math>}} of real numbers is not idempotent, but  {{nowrap|<math>-(\cdot)\circ -(\cdot)</math>}} is. In both cases, the composition is simply the [[identity function]], which is idempotent.