Category:Isomorphisms

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.

• 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.

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

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