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# 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.

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.

In sociology, isomorphism refers to a kind of "copying" or "imitation", especially of the practices of one organization by another.

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