homomorphism

English

English Wikipedia has an article on:

homomorphism

Wikipedia

Etymology

From homo- + morphism.

Pronunciation

Audio (US): (file)

Noun

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 $h : A \to B$ is a collection of homomorphisms $h_{U} : A (U) \to B (U)$ 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 $Φ$ from $G$ to $H$ is called a Lie group homomorphism if (1) $Φ$ is a group homomorphism and (2) $Φ$ is continuous.
(biology) A similar appearance of two unrelated organisms or structures, as for example with fish and whales.
Coordinate term: homology

Usage notes

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.

Hyponyms

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

Derived terms

Related terms

morphism

Translations

notion in mathematics

Arabic: تَشَاكُل m (tašākul)
Armenian: հոմոմորֆիզմը (homomorfizmə)
Bulgarian: хомоморфизъм m (homomorfizăm)
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)
Hindi: समाकारिता (samākāritā)
Hungarian: homomorfizmus (hu)
Icelandic: mótun f, sammótun f
Irish: homamorfacht f
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
Spanish: homomorfismo (es) m
Swedish: homomorfi (sv) c, homomorfism (sv) c
Ukrainian: гомоморфізм m (homomorfizm)
Vietnamese: phép đồng cấu (vi)