Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Idempotence
Summary
Idempotence (UK,ɪdɛmˈpəʊtəns, USˈaɪdəm-) 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 of a set equipped with a binary operator is said to be idempotent under if
The binary operation is said to be idempotent if
for all .
In the monoid of the natural numbers with multiplication, only 0 and 1 are idempotent. Indeed, and .
In the monoid +) of the natural numbers with addition, only 0 is idempotent. Indeed, 0 + 0 = 0.
In a magma , an identity element or an absorbing element , if it exists, is idempotent. Indeed, and .
In a group , the identity element is the only idempotent element. Indeed, if is an element of such that , then and finally by multiplying on the left by the inverse element of .
In the monoids and of the power set of the set with set union and set intersection respectively, and are idempotent. Indeed, for all , and for all .
In the monoids and of the Boolean domain with logical disjunction and logical conjunction respectively, and are idempotent. Indeed, for all , and for all .
In a GCD domain (for instance in ), the operations of GCD and LCM are idempotent.
In a Boolean ring, multiplication is idempotent.
In a Tropical semiring, addition is idempotent.
In a 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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.