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Encyclopedia > Idempotent

In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. There are two main definitions. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

  • Given a binary operation, an idempotent element (or simply an idempotent) is something that when multiplied by (for a function, composed with) itself, gives itself as a result. For example, the only two real numbers which are idempotent under multiplication are 0 and 1.
  • A unary operation (i.e., a function), is idempotent if, whenever it is applied twice to any element, it gives the same result as if it were applied once. For example, the greatest integer function is idempotent as a function from the set of real numbers to the set of integers.

Contents

In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a function returns a unique output for a given input. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...


Definition

Binary operation

Formally, if S is a set with a binary operation * on it, then an element s of S is said to be idempotent (with respect to *) if In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...

s * s = s.

In particular, any identity element is idempotent. If every element of S is idempotent, then the binary operation * is said to be idempotent. For example, the operations of set union and set intersection are both idempotent. 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 set. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... 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, all elements of B that also belong to A), but no other elements. ...


Unary operation

Formally, if f is a unary operation, say f maps X into Y, and if Y is a subset of X, then f is idempotent if, for all x in X, In mathematics, a unary operation is an operation with only one operand. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

f(f(x)) = f(x).

In particular, the identity function is idempotent, and any constant function is idempotent as well. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ...


Note that if X = Y, then we may consider S, the set of all functions from X to itself. In this case, function composition (denoted "o") is a binary operation on X, and a function f : XX is idempotent as a unary operator if and only if f o f = f, that is, if and only if f is an idempotent element of this binary operation. We say that f is idempotent on X. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... ↔ ⇔ ≡ logical symbols representing iff. ...


Common examples

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 greatest integer function of a real argument. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ... In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ...


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. It is an example of a closure operator; all closure operators are idempotent functions. Topological spaces are structures that allow one to formalize 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. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ...


Idempotent ring elements

An idempotent element of a ring is by definition an element that's idempotent with respect to thyukykye ring's multiplication. One may define a partial order on the idempotents of a ring as follows: if e and f are idempotents, we write ef iff ef = fe = e. With respect to this order, 0 is the smallest and 1 the largest idempotent. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...


If e is idempotent in the ring R, then eRe is again a ring, with multiplicative identity e.


Two idempotents e and f are called orthogonal if ef = fe = 0. In this case, e + f is also idempotent, and we have ee + f and fe + f.


If e is idempotent in the ring R, then so is f = 1 − e; e and f are orthogonal.


An idempotent e in R is called central if ex = xe for all x in R. In this case, Re is a ring with multiplicative identity e. The central idempotents of R are closely related to the decompositions of R as a direct sum of rings. If R is the direct sum of the rings R1,...,Rn, then the identity elements of the rings Ri are central idempotents in R, pairwise orthogonal, and their sum is 1. Conversely, given central idempotents e1,...,en in R which are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Re1,...,Ren. So in particular, every central idempotent e in R gives rise to a decomposition of R as a direct sum of Re and R(1 − e). In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...


Any idempotent e which is different from 0 and 1 is a zero divisor (because e(1 − e) = 0). This shows that integral domains and division rings don't have such idempotents. Local rings also don't have such idempotents, but for a different reason. The only idempotent that's contained in the Jacobson radical of a ring is 0. There is a catenoid of idempotents in the coquaternion ring. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. It is denoted by J(R) and can be defined in the following equivalent ways: the... A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. ... In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ...


A ring in which all elements are idempotent is called a boolean ring. It can be shown that in every such ring, multiplication is commutative, and every element is its own additive inverse. In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...


Other examples

Idempotent operations can be found in Boolean algebra as well. Logical and and logical or are both idempotent operations over the elements of the Boolean algebra. Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ... OR logic gate In mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...


In linear algebra, projections are idempotent. That is, any linear transformation that projects all vectors onto a subspace V (not necessarily orthogonally) is idempotent, if V itself is pointwise fixed. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ... In mathematics, a linear transformation (also called linear operator <<wrong! operators are LTs on the same vector space or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...


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 additive inverses. ...


See also

fixed point (mathematics) In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ...


  Results from FactBites:
 
Idempotent - definition of Idempotent in Encyclopedia (1030 words)
Given a binary operation, an idempotent element (or simply an idempotent) is something that when multiplied by (for a function, composed with) itself, gives itself as a result.
An idempotent element of a ring is by definition an element that's idempotent with respect to the ring's multiplication.
Idempotent behaviour is the basis for the caching ability of the web; if the same request delivers the same response one can also work with a copy instead of requiring the original page.
  More results at FactBites »

 
 

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