kernel

Let $\Sigma$ be a fixed signature, and $\mathfrak{A}$ and $\mathfrak{B}$ be two structures for $\Sigma$. Given a homomorphism $f\colon\mathfrak{A}\to\mathfrak{B}$, the kernel of $f$ is the relation $\ker(f)$ on $A$ defined by

 $\langle a,a^{\prime}\rangle\in\ker(f)\Leftrightarrow f(a)=f(a^{\prime}).$

So defined, the kernel of $f$ is a congruence on $\mathfrak{A}$. If $\Sigma$ has a constant symbol 0, then the kernel of $f$ is often defined to be the preimage of $0^{\mathfrak{B}}$ under $f$. Under this definition, if $\{0^{\mathfrak{B}}\}$ is a substructure of $\mathfrak{B}$, then the kernel of $f$ is a substructure of $\mathfrak{A}$.

Title kernel Kernel1 2013-03-22 13:46:34 2013-03-22 13:46:34 almann (2526) almann (2526) 11 almann (2526) Definition msc 03C05 msc 03C07 Kernel KernelOfAGroupHomomorphism KernelOfALinearTransformation