different between automorphism vs morphism

English

Etymology

auto- +? morphism

Noun

automorphism (plural automorphisms)

1. (algebra) An isomorphism of a mathematical object or system of objects onto itself.
2. The ascription to others of one's own characteristics.

Usage notes

• (algebra):
  • An automorphism is characterised by the structure it preserves, which is usually specified as an object type. Thus one may speak of a group automorphism or ring automorphism.
  • The identity mapping is sometimes called the trivial automorphism; any other automorphism may then be called a nontrivial automorphism.

Synonyms

• (isomorphism of a mathematical object or system of objects onto itself): self-map
• (ascription to others of one's own characteristics): projection

Hypernyms

• (algebra): isomorphism, endomorphism

Hyponyms

• (algebra): inner automorphism, outer automorphism, Möbius transformation

Derived terms

• automorphism group

Translations

See also

• self-map

Further reading

• Automorphism in the Encyclopædia Britannica (11th edition, 1911)

English

Etymology

Generalised from isomorphism, etc.

Noun

morphism (plural morphisms)

1. (mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.
   • 1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. ?encov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50,

     1° The composition of two morphisms is defined if and only if the final object of the first morphism is the initial object of the second. This composition is also a morphism, whose initial object is the initial object of the first morphism and whose final object is the final object of the second.

   • 1992, Terrance Brown (translator), Gil Henriques, Chapter 13: Morphisms and Transformations in the Construction of Invariants, Terrance Brown (translator), Jean Piaget, Gil Henriques, Edgar Ascher (editors), Morphisms and Categories: Comparing and Transforming, page 198,

     In certain extreme cases in mathematics, the synthesis of morphisms and of transformations is so intimate that one can speak of a veritable fusion. […] Essentially, categories are sets of morphisms organized into operatory systems.

   • 2007 November, Steven Dale Cutkosky, Toroidalization of Dominant Morphisms of 3-Folds, Memoirs of the American Mathematical Society, Volume 190, Number 890, page 3,

     The proof of toroidalization of morphisms of 3-folds to surfaces in [C3] breaks up into two parts: a reduction to prepared morphisms and then a proof of toroidalization of prepared morphisms from n-folds to [surfaces] in [CK].

2. (biology) Being or having distinct variants of a plant or animal species in the same locale; polymorphism.

Synonyms

• (category theory): arrow, map

Derived terms

• bimorphism
• epimorphism
• monomorphism

Related terms

• automorphism
• diffeomorphism
• homeomorphism
• homomorphism
• isomorphism

Translations

See also

• bijection
• function
• functor
• isometry
• measurable function