Алгебрадағы идемпотенттік қасиеті

Property of operations
the concepts in algebra

Idempotence (/UK//,//ɪ//d//ɛ//m/'/p//əʊ//t//ən//s/, /US/'/aɪ//d//ə//m// /) 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).

In the monoid of the natural numbers with multiplication, only and are idempotent. Indeed, 1= and 1=. In the monoid of the natural numbers with addition, only is idempotent. Indeed, 1=0 + 0 = 0. In a magma , an identity element or an absorbing element , if it exists, is idempotent. Indeed, 1= and 1=. In a group , the identity element is the only idempotent element. Indeed, if is an element of such that 1=, then 1= 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, 1= for all , and 1= for all In the monoids and of the Boolean domain with logical disjunction and logical conjunction respectively, and are idempotent. Indeed, 1= for all , and 1= 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.

Applied examples that many people could encounter in their day to day lives include elevator call buttons and crosswalk buttons. The initial activation of the button moves the system into a requesting state, until the request is satisfied. Subsequent activations of the button between the initial activation and the request being satisfied have no effect, unless the system is designed to adjust the time for satisfying the request based on the number of activations.