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 unknown 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 question are groups, such an isomorphism is called a group isomorphism. Similarly, if the objects are fields, it is called a field 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 graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the "edge structure" in the sense that there is an edge from vertex u to vertex v in G iff there is an edge from f(u) to f(v) in H.

See also

Isomorphism class, Homomorphism, Morphism

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