# Category:Definitions/Isomorphisms

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This category contains definitions related to Isomorphisms.

Related results can be found in Category:Isomorphisms.

- 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.

- Relation isomorphism: An

- 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.

- An

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

- Isomorphism (Hilbert Spaces): An

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

## Pages in category "Definitions/Isomorphisms"

The following 5 pages are in this category, out of 5 total.