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 of Σ,

  3. 3.

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


Homomorphisms with various additional properties have special names:

  • An injectivePlanetmathPlanetmath ( 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,


    the converseMathworldPlanetmath of condition 3 above, holds.

  • A surjective embedding is called an isomorphismMathworldPlanetmathPlanetmath.

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

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

Title structure homomorphism
Canonical name StructureHomomorphism
Date of creation 2013-03-22 12:43:22
Last modified on 2013-03-22 12:43:22
Owner almann (2526)
Last modified by almann (2526)
Numerical id 14
Author almann (2526)
Entry type Definition
Classification msc 03C07
Synonym homomorphism
Synonym morphism
Synonym monomorphism
Synonym epimorphism
Synonym bimorphism
Synonym embedding
Synonym isomorphism
Synonym endomorphism
Synonym automorphism
Related topic AxiomaticTheoryOfSupercategories
Defines structure morphism
Defines structure monomorphism
Defines structure epimorphism
Defines structure bimorphism
Defines structure embedding
Defines structure isomorphism
Defines structure endomorphism
Defines structure automorphism