英和和英

語源

From homo- +‎ morphism.

発音

名詞

homomorphism (plural homomorphisms)

1. (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 ${\displaystyle h:A\rightarrow B}$ 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, as for example with fish and whales.

   Coordinate term: homology

使用する際の注意点

In mathematics, this word is largely interchangeable with morphism. However, one uses morphism almost exclusively in the context of category theory, while homomorphism is more common in algebra.

下位語

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

派生語

関連する語

• morphism

参考

• (biology): convergent evolution

Further reading

• Group homomorphism on Wikipedia.Wikipedia
• Ring homomorphism on Wikipedia.Wikipedia
• Linear map on Wikipedia.Wikipedia
• Algebra homomorphism on Wikipedia.Wikipedia
• Glossary of field theory on Wikipedia.Wikipedia
• Semigroup on Wikipedia.Wikipedia
• Homomorphism on Encyclopedia of Mathematics
• homomorphism on nLab