Congruence relation
A congruence relation is to an equivalence relation what a homomorphism is to a function: it is an equivalence relation which somehow respects the algebraic structure of the set being partitioned; i.e. it is structure-preserving. Indeed, congruence relations correspond exactly to equivalence relations induced by a homomorphism.
Due to the structure-preserving property, a congruence relation defines a new algebraic structure on the equivalence classes under the relation, known as the Algebraic quotient.
Examples
Group congruence relation
Given a group
Properties
- Due to the correspondence between normal subgroups and congruence relations, congruence relations are usually represented by a normal subgroup, especially in constructions like the Quotient group.
Category congruence relation
Given a category