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Difference between revisions of "Kernel congruence"

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''of a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552201.png" /> of algebraic systems''
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''of a homomorphism $\phi : A \rightarrow A'$ of algebraic systems''
  
The congruence (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552202.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552203.png" /> consisting of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552204.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552205.png" />. For any congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552206.png" /> on an algebraic system there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552207.png" /> of this system for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552208.png" /> is the kernel congruence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k0552209.png" /> is the kernel congruence of a strong homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522010.png" /> of an algebraic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522011.png" /> onto a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522012.png" />, then the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522014.png" />, is an isomorphism of the quotient system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522015.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055220/k05522016.png" />.
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The congruence (cf. [[Congruence (in algebra)]]) $\theta$ on $A$ consisting of all pairs $(a,b) \in A \times A$ for which $\phi(a) = \phi(b)$. 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|Homomorphism]].
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For references see [[Homomorphism]].
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Revision as of 23:08, 12 November 2016

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 $\phi(a) = \phi(b)$. 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=11822
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article