PlanetMath: kernel

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kernel

(Definition)

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 is the relation $\ker(f)$ on defined by

$\displaystyle \langle a,a'\rangle \in \ker(f) \Iff f(a) = f(a').$

So defined, the kernel of

is a congruence on $\mathfrak{A}$ . If $\Sigma$ has a constant symbol 0, then the kernel of

is often defined to be the preimage of $0^\mathfrak{B}$ under

. Under this definition, if $\{0^\mathfrak{B}\}$ is a substructure of $\mathfrak{B}$ , then the kernel of

is a substructure of $\mathfrak{A}$ .

"kernel" is owned by almann.

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See Also: kernel, kernel, kernel of a linear mapping

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Cross-references: substructure, preimage, constant symbol, congruence, relation, homomorphism, structures, signature, fixed
There are 16 references to this entry.

This is version 8 of kernel, born on 2003-07-20, modified 2004-02-28.
Object id is 4485, canonical name is Kernel5.
Accessed 3166 times total.

Classification:

AMS MSC:	03C05 (Mathematical logic and foundations :: Model theory :: Equational classes, universal algebra)
	03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)

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