# 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 $\phi(a) = \phi(b)$. 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 $A'$. | + | 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 $A'$. |

For references see [[Homomorphism]]. | For references see [[Homomorphism]]. | ||

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## Revision as of 22:09, 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 $\phi(a) = \phi(b)$, 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://encyclopediaofmath.org/index.php?title=Kernel_congruence&oldid=39723