kernelKernel52003-07-20 18:35:392004-02-28 02:29:57Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\A}{\mathfrak{A}}
\newcommand{\B}{\mathfrak{B}}Let $\Sigma$ be a fixed signature, and $\A$ and $\B$ be two structures for $\Sigma$. Given a homomorphism $f\colon \A \to \B$, the \emph{kernel} of $f$ is the relation $\ker(f)$ on $A$ defined by
\[
\tuple{a,a'} \in \ker(f) \Iff f(a) = f(a').
\]
So defined, the kernel of $f$ is a congruence on $\A$. If $\Sigma$ has a constant symbol 0, then the kernel of $f$ is often defined to be the preimage of $0^\B$ under $f$. Under this definition, if $\set{0^\B}$ is a substructure of $\B$, then the kernel of $f$ is a substructure of $\A$.