Isomorphism

In mathematics, an isomorphism ( AltGr ἴσος (ISOs ) -. "Equal" and μορφή ( morphé ) - " form ", " shape " ) is a mapping between two mathematical structures, the parts of a structure to " meaningful same" parts of a other structure bijective ( bijective ) are mapped.

Definition

Universal Algebra

In universal algebra, ie a function between two algebraic structures (eg groups, rings, vector spaces or bodies ) is an isomorphism if:

• Is bijective,
• Is a homomorphism.

Is there an isomorphism between two algebraic structures, then the two structures are called isomorphic to each other. Isomorphic structures are in some way " the same", namely when one disregards the representation of the elements of the underlying quantities, and the name of the relations and links.

The statement " and are isomorphic " is usually recorded by or through.

Is a bijective homomorphism between two algebraic structures, then it is always a bijective homomorphism. However, this does not apply to all mathematical structures must therefore be a general definition, which also owns other mathematical structures validity, additionally require that also

• Is a homomorphism.

Category theory

In category theory one defines an isomorphism in general as a morphism of a two-sided inverse has:

Special cases of this Isomorphiebegriffes example, are homeomorphisms as isomorphisms in the category of topological spaces and continuous maps or homotopy equivalences as isomorphisms in the category of topological spaces with the homotopy classes of maps as morphisms.

Functional Analysis

In the functional analysis is called a mapping between normed spaces is an isomorphism if it has the following properties:

• Is linear
• Is continuous
• The inverse function is also continuous

If additionally for all, it is called an isometric isomorphism.

Importance

In category theory is crucial that functors preserve isomorphisms, h d is an isomorphism in a category and a functor, then

Also an isomorphism in the category. In algebraic topology, this property is often exploited to distinguish spaces can: For example, if the fundamental groups of two spaces are not isomorphic, so the spaces are not homeomorphic.

Examples

Are quantities and with a binary operation, then is an isomorphism from to a bijection with

For everyone. For instance, the logarithm is an isomorphism from to, there.

Are the structures of groups, it means such a group isomorphism isomorphism. Usually one thinks of such isomorphisms between algebraic structures such as groups, rings, vector spaces or bodies.

Are and totally ordered sets, then an isomorphism from X to Y is an order-preserving bijection. These isomorphisms play an important role in the theory of ordinal numbers.

Are and metric spaces and f is an isomorphism from to the property

Then f is called an isometric isomorphism.

If we let in the examples given away the requirement of bijectivity, each obtained homomorphisms.