Idempotence

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This article is about the mathematical concept of idempotence. For the related computing concept, see Idempotence (computer science).

In mathematics, the concept of idempotence (IPA: [ˈaɪdɪmpoʊtəns]), which roughly means that some operation yields the same result whether it is done only once or several times, occurs in several places in abstract algebra, in particular in the theory of projectors and of closure operators.

There are two main definitions of idempotence in use:

Given a binary operation, an idempotent element (or simply an idempotent) is something that when multiplied by (or for a function, composed with) itself, gives itself as a result. For example, the only two 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. This "unary operation definition" is in fact a special case of the "binary operation definition", see below.

1 Formal definitions
- 1.1 Binary operation
- 1.2 Unary operation
2 Common examples
3 See also

[edit] Formal definitions

[edit] Binary operation

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

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.

[edit] Unary operation

If f is a unary operation, i.e. a map f from some set X into itself, then f is idempotent if, for all x in X,

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

This is equivalent to say that f o f = f, using function composition denoted by "o". Thus an idempotent unary operation on X is an idempotent element of the set X^X of all functions from X into itself, with respect to the binary operation "o", in the sense of the previous definition.

In particular, the identity function is idempotent, and any constant function is idempotent as well.

[edit] Common examples

[edit] 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.

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.

[edit] Idempotent ring elements

An idempotent element of a ring is by definition an element that's idempotent with respect to the 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 e ≤ f iff ef = fe = e. With respect to this order, 0 is the smallest and 1 the largest idempotent.

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 e ≤ e + f and f ≤ e + 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,...,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 e₁,...,e_n in R which are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Re₁,...,Re_n. 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).

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.

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.