# Category:Isomorphisms

Jump to navigation
Jump to search

This category contains results about **Isomorphisms**.

Definitions specific to this category can be found in Definitions/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.

## Subcategories

This category has the following 4 subcategories, out of 4 total.