# Definition:Isomorphism

## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Isomorphism** may refer to:

- 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 $\left({S, \circ, \preceq}\right)$ 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.

## Also see

## Linguistic Note

The word **isomorphism** derives from the Greek **morphe** (* μορφή*) meaning

**form**or

**structure**, with the prefix

**iso-**meaning

**equal**.

Thus **isomorphism** means **equal structure**.