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.


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


 * Order Theory:
 * Order isomorphism: A bijection between two ordered sets which is order-preserving in both directions.
 * Ordered structure isomorphism: a bijection $\phi: S \to T$ from an ordered structure $\struct {S, \circ, \preceq}$ to another $\struct {T, *, \preccurlyeq}$ which is both an isomorphism from the structure $\struct {S, \circ}$ to the structure $\struct {T, *}$ and an order isomorphism from the ordered set $\struct {S, \preceq}$ to the ordered set $\struct {T, \preccurlyeq}$.


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


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


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


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

Also see

 * Definition:Homomorphism
 * Definition:Automorphism