# homomorphism

## English

English Wikipedia has an article on:
Wikipedia

### Noun

homomorphism (plural homomorphisms)

1. (algebra) A structure-preserving map between two algebraic structures of the same type, such as groups, rings, or vector spaces.
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 ${\displaystyle h:A\rightarrow B}$ is a collection of homomorphisms ${\displaystyle h_{U}:A(U)\rightarrow B(U)}$ commuting with restrictions.
• 2003, Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, page 17,
Definition 1.15. Let ${\displaystyle G}$ and ${\displaystyle H}$ be matrix Lie groups. A map ${\displaystyle \Phi }$ from ${\displaystyle G}$ to ${\displaystyle H}$ is called a Lie group homomorphism if (1) ${\displaystyle \Phi }$ is a group homomorphism and (2) ${\displaystyle \Phi }$ is continuous.
2. (biology) A similar appearance of two unrelated organisms or structures.