Kernel congruence

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.