Isomorphism

An Isomorphism is a morphism that admits an inverse.

• Context:
  • It can be produced by an Isomorphism Task.
• Example(s):
  • a Graph Isomorphism.
• See: Mathematics, Homomorphism, Morphism, Mathematical Object, Automorphism, Algebraic Structure, Group (Mathematics), Ring (Mathematics), Bijective, Topology.

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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. 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.

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