structure homomorphism

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:

1. 1.

For every constant symbol $c$ of $\Sigma$, $f(c^{\mathfrak{A}})=c^{\mathfrak{B}}$.

2. 2.

For every natural number $n$ and every $n$-ary function symbol $F$ of $\Sigma$,

 $f(F^{\mathfrak{A}}(a_{1},...,a_{n}))=F^{\mathfrak{B}}(f(a_{1}),...,f(a_{n})).$
3. 3.

For every natural number $n$ and every $n$-ary relation symbol $R$ of $\Sigma$,

 $R^{\mathfrak{A}}(a_{1},\ldots,a_{n})\Rightarrow R^{\mathfrak{B}}(f(a_{1}),% \ldots,f(a_{n})).$

Homomorphisms with various additional properties have special names:

• An injective (http://planetmath.org/Injective) homomorphism is called a monomorphism.

• A surjective homomorphism is called an epimorphism.

• A bijective homomorphism is called a bimorphism.

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

 $R^{\mathfrak{B}}(f(a_{1}),\ldots,f(a_{n}))\Rightarrow R^{\mathfrak{A}}(a_{1},% \ldots,a_{n}),$

the converse of condition 3 above, holds.

• A surjective embedding is called an isomorphism.

• A homomorphism from a structure to itself (e.g. (http://planetmath.org/Eg), $f\colon\mathfrak{A}\to\mathfrak{A}$) 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