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单词 StructureHomomorphism
释义

structure homomorphism


Let Σ be a fixed signaturePlanetmathPlanetmathPlanetmath, and 𝔄 and 𝔅 be two structuresMathworldPlanetmath for Σ. The interesting functions from 𝔄 to 𝔅 are the ones that preserve the structure.

A function f:𝔄𝔅 is said to be a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (or simply morphismMathworldPlanetmath) if and only if:

  1. 1.

    For every constant symbol c of Σ, f(c𝔄)=c𝔅.

  2. 2.

    For every natural numberMathworldPlanetmath n and every n-ary function symbol F ofΣ,

    f(F𝔄(a1,,an))=F𝔅(f(a1),,f(an)).
  3. 3.

    For every natural number n and every n-ary relation symbol Rof Σ,

    R𝔄(a1,,an)R𝔅(f(a1),,f(an)).

Homomorphisms with various additional properties have special names:

  • An injectivePlanetmathPlanetmath (http://planetmath.org/Injective) homomorphism is called a monomorphismMathworldPlanetmathPlanetmath.

  • A surjectivePlanetmathPlanetmath homomorphism is called an epimorphismMathworldPlanetmath.

  • A bijectiveMathworldPlanetmath homomorphism is called a bimorphismPlanetmathPlanetmath.

  • An injective homomorphism f is called an embedding if, for every natural number n and every n-ary relation symbol R of Σ,

    R𝔅(f(a1),,f(an))R𝔄(a1,,an),

    the converseMathworldPlanetmath of condition 3 above, holds.

  • A surjective embedding is called an isomorphismMathworldPlanetmathPlanetmath.

  • A homomorphism from a structure to itself (e.g. (http://planetmath.org/Eg), f:𝔄𝔄) is called an .

  • An isomorphism from a structure to itself is called an automorphism.

Titlestructure homomorphism
Canonical nameStructureHomomorphism
Date of creation2013-03-22 12:43:22
Last modified on2013-03-22 12:43:22
Owneralmann (2526)
Last modified byalmann (2526)
Numerical id14
Authoralmann (2526)
Entry typeDefinition
Classificationmsc 03C07
Synonymhomomorphism
Synonymmorphism
Synonymmonomorphism
Synonymepimorphism
Synonymbimorphism
Synonymembedding
Synonymisomorphism
Synonymendomorphism
Synonymautomorphism
Related topicAxiomaticTheoryOfSupercategories
Definesstructure morphism
Definesstructure monomorphism
Definesstructure epimorphism
Definesstructure bimorphism
Definesstructure embedding
Definesstructure isomorphism
Definesstructure endomorphism
Definesstructure automorphism
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