| Version 7 |
Version 6 |
| \PMlinkescapeword{embedding} |
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| 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. |
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. |
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A function $f\colon \A \to \B$ is said to be a \emph{homomorphism} (or simply \emph{morphism}) if and only if:
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A function $f\colon \A \to \B$ is said to be a \emph{homomorphism} (or simply morphism) if and only if:
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| \begin{enumerate} |
\begin{enumerate} |
| \item For every constant symbol $c$ of $\Sigma$, $f(c^\A)=c^\B$. |
\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 |
\item For every natural number $n$ and every $n$-ary function symbol $F$ of |
| $\Sigma$, |
$\Sigma$, |
| \[ |
\[ |
| f(F^\A(a_1,...,a_n))=F^\B(f(a_1),...,f(a_n)). |
f(F^\A(a_1,...,a_n))=F^\B(f(a_1),...,f(a_n)). |
| \] |
\] |
| \item For every natural number $n$ and every $n$-ary relation symbol $R$ |
\item For every natural number $n$ and every $n$-ary relation symbol $R$ |
| of $\Sigma$, |
of $\Sigma$, |
| \[ |
\[ |
| R^\A(a_1, \ldots ,a_n) \Implies R^\B(f(a_1), \ldots,f(a_n)). |
R^\A(a_1, \ldots ,a_n) \Implies R^\B(f(a_1), \ldots,f(a_n)). |
| \] |
\] |
| \end{enumerate} |
\end{enumerate} |
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| Homomorphisms with various additional properties have special names: |
Homomorphisms with various additional properties have special names: |
| \begin{itemize} |
\begin{itemize} |
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\item An \PMlinkname{injective}{Injective} homomorphism is called a \emph{monomorphism}.
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\item An injective homomorphism is called a \emph{monomorphism}.
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| \item A surjective homomorphism is called an \emph{epimorphism}. |
\item A surjective homomorphism is called an \emph{epimorphism}. |
| \item A bijective homomorphism is called a \emph{bimorphism}. |
\item A bijective homomorphism is called a \emph{bimorphism}. |
| \item An injective homomorphism whose inverse function is also a homomorphism is called an \emph{embedding}. |
\item An injective homomorphism whose inverse function is also a homomorphism is called an \emph{embedding}. |
| \item A surjective embedding is called an \emph{isomorphism}. |
\item A surjective embedding is called an \emph{isomorphism}. |
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\item A homomorphism from a structure to itself (\PMlinkname{e.g.}{Eg}, $f\colon \A \to \A$) is called an \emph{\PMlinkescapetext{endomorphism}}.
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\item A homomorphism from a structure to itself (e.g., $f\colon \A \to \A$) is called an \emph{endomorphism}.
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| \item An isomorphism from a structure to itself is called an \emph{automorphism}. |
\item An isomorphism from a structure to itself is called an \emph{automorphism}. |
| \end{itemize} |
\end{itemize} |
