structure homomorphismStructurePreservingMappings2002-06-03 22:49:062007-11-14 16:23:22Definitionstructure morphismstructure monomorphismstructure epimorphismstructure bimorphismstructure embeddingstructure isomorphismstructure endomorphismstructure automorphism% this is the default PlanetMath preamble. as your knowledge
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\end{array}\right)}\PMlinkescapeword{embedding}
Let $\Sigma$ be a fixed signature, and $\A$ and $\B$ be two structures for $\Sigma$. The interesting functions from $\A$ to $\B$ are the ones that preserve the structure.
A function $f\colon \A \to \B$ is said to be a \emph{homomorphism} (or simply \emph{morphism}) if and only if:
\begin{enumerate}
\item For every constant symbol $c$ of $\Sigma$, $f(c^\A)=c^\B$.
\item For every natural number $n$ and every $n$-ary function symbol $F$ of
$\Sigma$,
\[
f(F^\A(a_1,...,a_n))=F^\B(f(a_1),...,f(a_n)).
\]
\item For every natural number $n$ and every $n$-ary relation symbol $R$
of $\Sigma$,
\[
R^\A(a_1, \ldots ,a_n) \Implies R^\B(f(a_1), \ldots,f(a_n)).
\]
\end{enumerate}
Homomorphisms with various additional properties have special names:
\begin{itemize}
\item An \PMlinkname{injective}{Injective} homomorphism is called a \emph{monomorphism}.
\item A surjective homomorphism is called an \emph{epimorphism}.
\item A bijective homomorphism is called a \emph{bimorphism}.
\item An injective homomorphism $f$ is called an \emph{embedding} if, for every natural number $n$ and every $n$-ary relation symbol $R$ of $\Sigma$,
\[
R^\B(f(a_1), \ldots,f(a_n)) \Implies R^\A(a_1, \ldots ,a_n),
\]
the converse of condition 3 above, holds.
\item A surjective embedding is called an \emph{isomorphism}.
\item A homomorphism from a structure to itself (\PMlinkname{e.g.}{Eg}, $f\colon \A \to \A$) is called an \emph{\PMlinkescapetext{endomorphism}}.
\item An isomorphism from a structure to itself is called an \emph{automorphism}.
\end{itemize}