Definition:Isomorphism

Disambiguation

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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.

• Order Theory:
  • Order isomorphism: A bijection between two posets 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 $\left({T, *, \preccurlyeq}\right)$ an isomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$ and an order isomorphism from the poset $\left({S, \preceq}\right)$ to the poset $\left({T, \preccurlyeq}\right)$.

• 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 (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.