homomorphism

English

homomorphism (plural homomorphisms)

(algebra) A structure-preserving map between two algebraic structures of the same type, such as groups, rings, or vector spaces.
Coordinate term: homology

A field homomorphism is a map from one field to another one which is additive, multiplicative, zero-preserving, and unit-preserving.
- 1954, Kuo-Tsai Chen, Iterated Integrals and Exponential Homomorphisms, Proceedings of the London Mathematical Society, Reprinted in 2001, Philippe Tondeur (editor), Collected Papers of K.-T. Chen, Birkhäuser, page 54,
  This motivates a generalization, and exponential homomorphisms are now defined, in an algebraic fashion, from certain free products to formal power series rings with non-commutative indeterminates.
- 1997, Glen E. Bredon, Sheaf Theory, 2nd edition, Springer, page 8:
  A homomorphism of presheaves $h:A\rightarrow B$ is a collection of homomorphisms $h_{U}:A(U)\rightarrow B(U)$ commuting with restrictions.
- 2003, Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, page 17:
  Definition 1.15. Let $G$ and $H$ be matrix Lie groups. A map $\Phi$ from $G$ to $H$ is called a Lie group homomorphism if (1) $\Phi$ is a group homomorphism and (2) $\Phi$ is continuous.
(biology) A similar appearance of two unrelated organisms or structures, as for example with fish and whales.
Coordinate term: homology

(structure-preserving map between algebraic structures): automorphism, endomorphism, epimorphism, isomorphism, linear map, monomorphism

notion in mathematics