homomorphism

From homo- +‎ morphism.

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homomorphism (plural homomorphisms)

2. (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.

4. (biology) A similar appearance of two unrelated organisms or structures, as for example with fish and whales.

   Coordinate term: homology

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.

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

• morphism

• (biology): convergent evolution

• 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