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