PlanetMath: structure homomorphism

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structure homomorphism

(Definition)

Let $\Sigma$ be a fixed signature, and $\mathfrak{A}$ and $\mathfrak{B}$ be two structures for $\Sigma$ . The interesting functions from $\mathfrak{A}$ to $\mathfrak{B}$ are the ones that preserve the structure.

A function $f\colon \mathfrak{A}\to \mathfrak{B}$ is said to be a homomorphism (or simply morphism) if and only if:

For every constant symbol of $\Sigma$ , $f(c^\mathfrak{A})=c^\mathfrak{B}$ .
For every natural number and every -ary function symbol of $\Sigma$ ,
$\displaystyle f(F^\mathfrak{A}(a_1,...,a_n))=F^\mathfrak{B}(f(a_1),...,f(a_n)).$
For every natural number and every -ary relation symbol of $\Sigma$ ,
$\displaystyle R^\mathfrak{A}(a_1, \ldots ,a_n) \Rightarrow R^\mathfrak{B}(f(a_1), \ldots,f(a_n)).$

Homomorphisms with various additional properties have special names:

An injective homomorphism is called a monomorphism.
A surjective homomorphism is called an epimorphism.
A bijective homomorphism is called a bimorphism.
An injective homomorphism is called an embedding if, for every natural number and every -ary relation symbol of $\Sigma$ ,
$\displaystyle R^\mathfrak{B}(f(a_1), \ldots,f(a_n)) \Rightarrow R^\mathfrak{A}(a_1, \ldots ,a_n),$
the converse of condition 3 above, holds.
A surjective embedding is called an isomorphism.
A homomorphism from a structure to itself (e.g., $f\colon \mathfrak{A}\to \mathfrak{A}$ ) is called an endomorphism.
An isomorphism from a structure to itself is called an automorphism.

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"structure homomorphism" is owned by almann. [ full author list (6) | owner history (2) ]

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Other names:

homomorphism, morphism, monomorphism, epimorphism, bimorphism, embedding, isomorphism, endomorphism, automorphism

Also defines:

structure morphism, structure monomorphism, structure epimorphism, structure bimorphism, structure embedding, structure isomorphism, structure endomorphism, structure automorphism

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Cross-references: converse, injective, bijective, surjective, properties, relation symbol, function symbol, natural number, constant symbol, preserve, functions, structures, signature, fixed
There are 117 references to this entry.

This is version 11 of structure homomorphism, born on 2002-06-03, modified 2007-11-14.
Object id is 3021, canonical name is StructurePreservingMappings.
Accessed 22182 times total.

Classification:

AMS MSC:

03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)

Pending Errata and Addenda

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