different between isomorphism vs automorphism

English

Etymology

iso- +? -morphism

Pronunciation

Noun

isomorphism (plural isomorphisms)

1. Similarity of form
   • 1984 Brigitte Asbach-Schnitker, "Introduction", Mercury or The Secret and Swift Messenger, ?ISBN.

     The postulated isomorphism between words and things constitutes the characterizing feature of all philosophically based universal languages.

   1. (biology) the similarity in form of organisms, which may be due to convergent evolution or shared genetic background, e.g. an algae species in which the haploid and diploid life stages are indistinguishable based on morphology.
   2. (chemistry) the similarity in the crystal structures of similar chemical compounds
      • 1874 C. Rammelsberg, "Crystallographic and chemical relations of the natural sulphides, arsenides, and sulpharsenides", The Chemical News and Journal of Physical Science, page 197.

        The isomorphism of compounds does not prove the isomorphism of their respective constituents.

   3. (sociology) the similarity in the structure or processes of different organizations
2. A one-to-one correspondence
   1. (group algebra) A bijection f such that both f and its inverse f ^?1 are homomorphisms, that is, structure-preserving mappings.
   2. (computer science) a one-to-one correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
   3. (category theory) A morphism which has an inverse; the composition of the morphism and its inverse yields either one of two identity morphisms (depending on the order of composition).

Synonyms

• (in category theory): iso

Antonyms

• anisomorphism

Related terms

• isomorphic
• isomorphous

Translations

See also

• Isomorphism on Wikipedia.Wikipedia
• Isomorphism on Encyclopedia of Mathematics
• isomorphism on nLab
• Isomorphism on Wolfram MathWorld
• Graph Isomorphism on Wolfram MathWorld
• Natural Isomorphism on Wolfram MathWorld

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)