Difference between revisions of "Kernel congruence"
From Encyclopedia of Mathematics
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| − | + | {{TEX|done}}{{MSC|08A30}} | |
| − | + | ''of a homomorphism $\phi : A \rightarrow A'$ of algebraic systems'' | |
| − | For references see [[ | + | 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 $A'$. |
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| + | For references see [[Homomorphism]]. | ||
Latest revision as of 07:39, 13 November 2016
2020 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://encyclopediaofmath.org/index.php?title=Kernel_congruence&oldid=11822
Kernel congruence. Encyclopedia of Mathematics. URL: http://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