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
Canonical name	Kernel1
Date of creation	2013-03-22 13:46:34
Last modified on	2013-03-22 13:46:34
Owner	almann (2526)
Last modified by	almann (2526)
Numerical id	11
Author	almann (2526)
Entry type	Definition
Classification	msc 03C05
Classification	msc 03C07
Related topic	Kernel
Related topic	KernelOfAGroupHomomorphism
Related topic	KernelOfALinearTransformation