Idempotence

Idempotence is a term from mathematics and computer science. In mathematics is called an object that a link has the property as idempotent with respect to this link. An important special case is idempotent with respect to the consecutive execution of features. Analogously, in the computer science a function after another called multiple times the same result as with a single command will be referred to as idempotent.

Definitions

Idempotent elements

An element of a set is called idempotent with respect to a - digit combination and if the following applies:

The other hand, satisfies the equation for a one-digit shortcut

Then is a fixed point of

Idempotent functions

This is called a one-digit shortcut or function idempotent if it is idempotent with respect to the composition:

I.e., for all the results in a double application of the same value as the one-time.

Idempotent algebraic structures

Are all elements of a semigroup with respect to idempotent, then it is also even called idempotent. Alternatively, an idempotent semigroup is also often referred to as a band. Every commutative band is called semilattice. This is called a semigroup globally idempotent if the following applies:

A half-ring a fast ring and a ring is called idempotent if each is or idempotent. In contrast, a Dioid is a Hemiring with identity and idempotent addition.

Examples

Idempotent operations:

• With respect to multiplication and the solutions of the equation are the only idempotent real numbers.
• For a two -digit shortcut is a (left or right ) neutral element always idempotent: In a group, the neutral element is the only idempotent element.
• In a ring with unit 0 and 1 are always idempotent elements. If the ring is not a body, even more idempotent elements can exist. For example, under the residue class ring

• Is an association, so are half and associations.

Idempotent pictures:

• Constant functions:
• Identical illustration:
• Projections, for example,
• Amount functions:
• Cover operators.
• Nuclear operators.

Properties

• A matrix on an arbitrary body if and idempotent with respect to the conventional matrix multiplication if they induced linear mapping

• Each idempotent ring is commutative, because it applies to all

• An idempotent near-ring is commutative if and only if it is distributive, because:

Computer science

In computer science, idempotency is required of recovery measures for databases and services in order to provide fault tolerance in a crash during a restart phase. Undo and redo operations must here have the same result result after multiple sequential execution.

Purely read services are idempotent nature, since the state of the data is not changed. Any non- idempotent writing service can be made ​​from a technical perspective to an idempotent service.

For a service to the posting of funds to deposit the call is (100 ) is not idempotent because after multiple service call, the amount is 100 times more paid up. If one, however, invoke new account was (600), it would remain the same after repeated service call, the account balance. This call would be idempotent.