Isomorphism
In mathematics, an isomorphism is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition:
- The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49)
Formally, an isomorphism is a bijective map f such that both f and its inverse f -1 are homomorphisms, i.e. structure-preserving mappings.
If there exists an isomorphism between two structures, we call the two structures isomorphic. Isomorphic structures are "the same" at a certain level of abstraction; ignoring the specific identities of the elements in the underlying sets and the names of the underlying relations, the two structures are identical.
For example, if one object consists of a set X with an ordering <= and the other object consists of a set Y with an ordering then an isomorphism from X to Y is a bijective function f : X -> Y such that
- f(u) [= f(v) iff u <= v.]
Such an isomorphism is called an order isomorphism.
Or, if on these sets the binary operations * and @ are defined, respectively, then an isomorphism from X to Y is a bijective function f : X -> Y such that
- f(u) @ f(v) = f(u * v)
for all u, v in X.
When the objects in questions are groups, such an isomorphism is called a group isomorphism.
In universal algebra, one can give a general definition of isomorphism that covers these and many other cases.
The definition of isomorphism given in category theory is even more general.
See also:
Isomorphism class, Homomorphism, Morphism
In sociology, isomorphism refers to a kind of "copying" or "imitation", especially of the practices of one organization by another.
Referenced By
AutoMorphism | List of mathematical topics (G-I) | List of mathematical topics (G-Z) | Type (logic) | Type theory | Typed logic
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