structure homomorphism
Let Σ be a fixed signature, and 𝔄 and 𝔅 be two structures for Σ. The interesting functions from 𝔄 to 𝔅 are the ones that preserve the structure.
A function f:𝔄→𝔅 is said to be a homomorphism (or simply morphism) if and only if:
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1.
For every constant symbol c of Σ, f(c𝔄)=c𝔅.
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2.
For every natural number n and every n-ary function symbol F of Σ,
f(F𝔄(a1,…,an))=F𝔅(f(a1),…,f(an)). -
3.
For every natural number n and every n-ary relation symbol R of Σ,
R𝔄(a1,…,an)⇒R𝔅(f(a1),…,f(an)).
Homomorphisms with various additional properties have special names:
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An injective (http://planetmath.org/Injective) homomorphism is called a monomorphism.
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A surjective homomorphism is called an epimorphism.
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A bijective homomorphism is called a bimorphism.
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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 converse of condition 3 above, holds.
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A surjective embedding is called an isomorphism.
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A homomorphism from a structure to itself (e.g. (http://planetmath.org/Eg), f:𝔄→𝔄) is called an .
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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 |