# Isomorphism

An Isomorphism is a morphism that admits an inverse.

**Context:**- It can be produced by an Isomorphism Task.

**Example(s):****See:**Mathematics, Homomorphism, Morphism, Mathematical Object, Automorphism, Algebraic Structure, Group (Mathematics), Ring (Mathematics), Bijective, Topology.

## References

### 2014

- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/isomorphism Retrieved:2014-4-24.
- In mathematics, an
**isomorphism**, from the Greek: ἴσος*isos*"equal", and μορφή*morphe*"shape", is a homomorphism (or more generally a morphism) that admits an inverse.^{[1]}Two mathematical objects are**isomorphic**if an isomorphism exists between them. An*automorphism*is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

In topology, where the morphisms are continuous functions, isomorphisms are also called

*homeomorphisms*or*bicontinuous functions*. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called*diffeomorphisms*.Isomorphisms are formalized using category theory. A morphism in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism in that category such that 1

_{X}and 1_{Y}, where 1_{X}and 1_{Y}are the identity morphisms of*X*and Y, respectively.^{[2]}

- In mathematics, an