isomorphism

In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as . The word is derived .

AI overview

An isomorphism is a special kind of mathematical relationship between two structures that preserves their essential organization and can be reversed, meaning you can go back and forth between the structures without losing information. When two mathematical structures are isomorphic, it means they have the same underlying form or pattern, even if they might look different on the surface, which helps mathematicians recognize that seemingly different problems or systems are actually equivalent in important ways.

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Article

16 sections

Contents

Examples
Logarithm and exponential
Integers modulo 6
Relation-preserving isomorphism
Applications
Category theoretic view
Isomorphism vs. bijective morphism
Isomorphism class
Examples
Relation to equality
Notation
See also
Notes
References
Further reading
External links

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified.