structure homomorphism
Let $\mathrm{\Sigma}$ be a fixed signature^{}, and $\U0001d504$ and $\U0001d505$ be two structures^{} for $\mathrm{\Sigma}$. The interesting functions from $\U0001d504$ to $\U0001d505$ are the ones that preserve the structure.
A function $f:\U0001d504\to \U0001d505$ is said to be a homomorphism^{} (or simply morphism^{}) if and only if:

1.
For every constant symbol $c$ of $\mathrm{\Sigma}$, $f({c}^{\U0001d504})={c}^{\U0001d505}$.

2.
For every natural number^{} $n$ and every $n$ary function symbol $F$ of $\mathrm{\Sigma}$,
$$f({F}^{\U0001d504}({a}_{1},\mathrm{\dots},{a}_{n}))={F}^{\U0001d505}(f({a}_{1}),\mathrm{\dots},f({a}_{n})).$$ 
3.
For every natural number $n$ and every $n$ary relation symbol $R$ of $\mathrm{\Sigma}$,
$${R}^{\U0001d504}({a}_{1},\mathrm{\dots},{a}_{n})\Rightarrow {R}^{\U0001d505}(f({a}_{1}),\mathrm{\dots},f({a}_{n})).$$
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 $\mathrm{\Sigma}$,
$${R}^{\U0001d505}(f({a}_{1}),\mathrm{\dots},f({a}_{n}))\Rightarrow {R}^{\U0001d504}({a}_{1},\mathrm{\dots},{a}_{n}),$$ 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), $f:\U0001d504\to \U0001d504$) is called an .

•
An isomorphism from a structure to itself is called an automorphism.
Title  structure homomorphism 
Canonical name  StructureHomomorphism 
Date of creation  20130322 12:43:22 
Last modified on  20130322 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 