Idempotence (UK: /ˌɪdɛmˈpoʊtəns/, US: /ˈaɪdəm-/) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).

The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power).

An element ${\displaystyle x}$ of a set ${\displaystyle S}$ equipped with a binary operator ${\displaystyle \cdot }$ is said to be idempotent under ${\displaystyle \cdot }$ if

The binary operation ${\displaystyle \cdot }$ is said to be idempotent if

In the monoid ${\displaystyle (E^{E},\circ )}$ of the functions from a set ${\displaystyle E}$ to itself (see set exponentiation) with function composition ${\displaystyle \circ }$, idempotent elements are the functions ${\displaystyle f\colon E\to E}$ such that ${\displaystyle f\circ f=f}$, that is such that ${\displaystyle f(f(x))=f(x)}$ for all ${\displaystyle x\in E}$ (in other words, the image ${\displaystyle f(x)}$ of each element ${\displaystyle x\in E}$ is a fixed point of ${\displaystyle f}$). For example:

If the set ${\displaystyle E}$ has ${\displaystyle n}$ elements, we can partition it into ${\displaystyle k}$ chosen fixed points and ${\displaystyle n-k}$ non-fixed points under ${\displaystyle f}$, and then ${\displaystyle k^{n-k}}$ is the number of different idempotent functions. Hence, taking into account all possible partitions,

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, ... (sequence A000248 in the OEIS).

Neither the property of being idempotent nor that of being not is preserved under function composition. As an example for the former, ${\displaystyle f(x)=x}$ mod 3 and ${\displaystyle g(x)=\max(x,5)}$ are both idempotent, but ${\displaystyle f\circ g}$ is not, although ${\displaystyle g\circ f}$ happens to be. As an example for the latter, the negation function ${\displaystyle \neg }$ on the Boolean domain is not idempotent, but ${\displaystyle \neg \circ \neg }$ is. Similarly, unary negation ${\displaystyle -(\cdot )}$ of real numbers is not idempotent, but ${\displaystyle -(\cdot )\circ -(\cdot )}$ is. In both cases, the composition is simply the identity function, which is idempotent.