Definition:Isomorphism


 * Isomorphism (Abstract Algebra): An isomorphism between two algebraic structures is a bijection which preserves operations.
 * Group isomorphism: an isomorphism between two groups.
 * Ring isomorphism: an isomorphism between two rings.
 * $R$-algebraic structure isomorphism: an isomorphism between two $R$-algebraic structures.


 * Isomorphism (Category Theory): A morphism $f: X \to Y$ for which if there exists a morphism $g: Y \to X$ such that $g \circ f = \operatorname{id}_X$ and $f \circ g = \operatorname{id}_Y$.


 * Isomorphism (Graph Theory): An isomorphism between two graphs is a bijection which preserves incidences between edges and vertices.


 * Isomorphism (Hilbert Spaces): An isomorphism between two Hilbert spaces is a linear surjection which preserves the inner product.


 * Relation isomorphism: A isomorphism between two relational structures is a bijection which preserves relations.


 * Order isomorphism: An (order) isomorphism between two posets is a bijection which preserves ordering.


 * Isomorphism (Topology): same thing as a homeomorphism.