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Difference between revisions of "Kernel congruence"

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''of a homomorphism $\phi : A \rightarrow A'$ of algebraic systems''
 
''of a homomorphism $\phi : A \rightarrow A'$ of algebraic systems''
  
The congruence (cf. [[Congruence (in algebra)]]) $\theta$ on $A$ consisting of all pairs $(a,b) \in A \times A$ for which $\phi(a) = \phi(b)$, cf. [[Kernel of a function]]. For any congruence $\theta$ on an algebraic system there is a homomorphism $\phi$ of this system for which $\theta$ is the kernel congruence. If $\theta$ is the kernel congruence of a strong homomorphism $\phi$ of an algebraic system $A$ onto a system $A'$, then the canonical mapping $a/\theta \mapsto \phi(a)$, where $a/\theta = \{ b \in A : (b,a) \in \theta \}$, is an isomorphism of the quotient system $A/\theta$ onto $A'$.
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The congruence (cf. [[Congruence (in algebra)]]) $\theta$ on an [[algebraic system]] $A$ consisting of all pairs $(a,b) \in A \times A$ for which $\phi(a) = \phi(b)$, cf. [[Kernel of a function]]. For any congruence $\theta$ on an algebraic system there is a homomorphism $\phi$ of this system for which $\theta$ is the kernel congruence. If $\theta$ is the kernel congruence of a [[strong homomorphism]] $\phi$ of an algebraic system $A$ onto a system $A'$, then the canonical mapping $a/\theta \mapsto \phi(a)$, where $a/\theta = \{ b \in A : (b,a) \in \theta \}$, is an isomorphism of the quotient system $A/\theta$ onto $A'$.
  
 
For references see [[Homomorphism]].
 
For references see [[Homomorphism]].
 
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Latest revision as of 08:39, 13 November 2016

2010 Mathematics Subject Classification: Primary: 08A30 [MSN][ZBL]

of a homomorphism $\phi : A \rightarrow A'$ of algebraic systems

The congruence (cf. Congruence (in algebra)) $\theta$ on an algebraic system $A$ consisting of all pairs $(a,b) \in A \times A$ for which $\phi(a) = \phi(b)$, cf. Kernel of a function. For any congruence $\theta$ on an algebraic system there is a homomorphism $\phi$ of this system for which $\theta$ is the kernel congruence. If $\theta$ is the kernel congruence of a strong homomorphism $\phi$ of an algebraic system $A$ onto a system $A'$, then the canonical mapping $a/\theta \mapsto \phi(a)$, where $a/\theta = \{ b \in A : (b,a) \in \theta \}$, is an isomorphism of the quotient system $A/\theta$ onto $A'$.

For references see Homomorphism.

How to Cite This Entry:
Kernel congruence. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kernel_congruence&oldid=39725
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article