left and right euclidean;
Since is a monomorphism, the maps , , and are necessarily unique if they exist.
Every kernel pair is a congruence.
A congruence which is the kernel pair of some morphism is called effective.
An effective congruence is always the kernel pair of its quotient if that quotient exists.
The quotient of an effective congruence is an effective quotient.
A special case of this is that of a quotient module.
The notions of regular category and exact category can naturally be formulated in terms of congruences. A “higher arity” version, corresponding to coherent categories and pretoposes is discussed at familial regularity and exactness.