Idempotence

Idempotence IPA: /ˌaɪdɨmˈpoʊtənts/ describes the property of operations in mathematics and computer science which yield the same result after the operation is applied multiple times. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their 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). Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and

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. In Mathematics, a unary operation is an operation with only one Operand, i The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function For example, the absolute value function is idempotent as a function from the set of real numbers to the set of real numbers: abs(abs x) = abs(x). In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
A binary operation is called idempotent if, whenever it is applied to two equal values, it gives that value as the result. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two For example, the operation giving the maximum value of two values is idempotent: max(x, x) = x.
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: 1 × 1 = 1.

1 Definitions
2 Common examples
3 See also

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 Mathematics, a unary operation is an operation with only one Operand, i

In particular, the identity function id_S, defined by id_S(x) = x, is idempotent, as is the constant function K_c, where c is an element of S, defined by K_c(x) = c. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that In Mathematics, a constant function is a function whose values do not vary and thus are Constant.

An important class of idempotent functions is given by projections in a vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added An example of a projection is the function π_xy defined by π_xy(x, y, z) = (x, y, 0), 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. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

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. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of In mathematics a meet on a set is defined either as the unique Infimum (greatest lower bound with respect to a Partial order on the set provided an infimum exists In mathematical Order theory, join is a Binary operation on a Partially ordered set that gives the Supremum (least upper bound of its arguments In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements'

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. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

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, f(f(x)) = f(x) for x in the domain of f, can equivalently be rewritten as f o f = f, using the binary operation of function composition denoted by "o". In Mathematics, a composite function represents the application of one function to the results of another 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

Main article: Referential transparency (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. Referential transparency and referential opaqueness are properties of parts of Computer programs An expression is said to be referentially transparent Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Object-oriented programming, the term method refers to a Subroutine that is exclusively associated either with a class (called class methods In Computer science, a subroutine ( function, method, procedure, or subprogram) is a portion of code within a larger 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. Event Stream Processing, or ESP, is a set of technologies designed to assist the construction of event-driven information systems.

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. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The

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. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S " In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) It is an example of a closure operator; all closure operators are idempotent functions. A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S

Idempotent ring elements

An idempotent element of a ring is, by definition, an element which is idempotent with respect to the ring's multiplication. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement ↔ 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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction If R is the direct sum of the rings R₁,. . . ,R_n, then the identity elements of the rings R_i are central idempotents in R, pairwise orthogonal, and their sum is 1. Conversely, given central idempotents a₁,. . . ,a_n in R which are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra₁,. . . ,Ra_n. 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). In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 This shows that integral domains and division rings don't have such idempotents. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible Local rings also don't have such idempotents, but for a different reason. In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called The only idempotent contained in the Jacobson radical of a ring is 0. In Ring theory, a branch of Abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements There is a catenoid of idempotents in the coquaternion ring. A catenoid is a three- Dimensional Shape made by rotating a Catenary Curve around the x axis In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the

A ring in which all elements are idempotent is called a boolean ring. In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that It can be shown that in every such ring, multiplication is commutative, and every element is its own additive inverse. In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero.

Relation with involutions

If a is an idempotent, then $1 - 2 a$ is an involution.

If b is an involution, then $\frac{1}{2}(1-b)$ 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 $\frac{1}{2}(1-b)$ and $\frac{1}{2}(1+b)$ are orthogonal idempotents, corresponding to a and $1 - a$ .

Numerical examples

One may consider the ring of integers mod n, where n is squarefree. In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra. 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 $22$ 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. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s

In linear algebra, projections are idempotent. Linear algebra is the branch of Mathematics concerned with In fact, they are defined as idempotent linear maps. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

An idempotent semiring is a semiring whose addition (not multiplication) is idempotent. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse

There are idempotent matrices.

Dictionary

-noun

(mathematics) (computing) a quality of an action such that repetitions of the action have no further effect on outcome.

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