View source for Kernel congruence

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''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]].