Isomorphism

In abstract algebra, an isomorphism is a bijective homomorphism. Two mathematical structures are said to be isomorphic if there is an isomorphism between them.

In essence, two objects are isomorphic if they are indistinguishable given only a selection of their features, and the isomorphism is the mapping of the set elements and the selected operations between the objects. A named isomorphism indicates which features are selected for this purpose. Thus, for example, two objects may be group isomorphic without being ring isomorphic, since the latter isomorphism selects the additional structure of the multiplicative operator.

In category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f −1f = id_X and f f −1 = id_Y.