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 $\left({S, \circ, \preceq}\right)$ to another $\left({T, *, \preccurlyeq}\right)$ which is both an isomorphism from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$ and an order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.


 * 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


 * 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.


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