homomorphism

English

Etymology

homo- +‎ morphism.

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 $h:A\rightarrow B$ 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.

2. (biology) A similar appearance of two unrelated organisms or structures.

Hyponyms

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

Derived terms

• ring homomorphism
• semigroup homomorphism

Related terms

• morphism

Translations

notion in mathematics

• Arabic: تَشَاكُل m (tašākul)
• Armenian: հոմոմորֆիզմը (homomorfizmə)
• Czech: homomorfismus (cs) m
• Danish: homomorfi c
• Dutch: homomorfisme (nl) n
• Finnish: homomorfismi
• French: homomorphisme (fr) m
• Georgian: ჰომომორფიზმი (homomorpizmi)
• German: Homomorphismus (de) m
• Greek: ομομορφισμός m (omomorfismós)
• Hungarian: homomorfizmus (hu)
• Icelandic: mótun f, sammótun f
• Irish: homamorfacht f
• (deprecated template usage) {{trans-mid}}
• Italian: omomorfismo (it) m
• Lithuanian: homomorfizmas
• Persian: همریختی (fa)
• Polish: homomorfizm m
• Portuguese: homomorfismo m
• Punjabi: ਹੋਮੋਮੌਰਫਿਜ਼ਮ (homomaurphizam)
• Russian: гомоморфи́зм (ru) m (gomomorfízm)
• Serbo-Croatian: homomorfizam (sh) m
• Slovene: homomorfizem m
• Swedish: homomorfi (sv) c, homomorfism (sv) c
• Ukrainian: гомоморфізм m (homomorfizm)
• Vietnamese: phép đồng cấu (vi)

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