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 $(𝐺, \cdot)$ then an Equivalence relation $\equiv$ is a congruence relation iff.

𝑔 1 \equiv 𝑔 2 \land ℎ 1 \equiv ℎ 2 ⟹ 𝑔 1 \cdot ℎ 1 \equiv 𝑔 2 \cdot ℎ 2

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 $𝖢$ then a a family of equivalence relations on every hom-set $\equiv$ is an equivalence relation iff. $𝑓 1 \equiv 𝑓 2 : 𝑋 \to 𝑌$ and $𝑔 1 \equiv 𝑔 2 : 𝑌 \to 𝑍$ implies $𝑔 1 𝑓 1 \equiv 𝑔 2 𝑓 2 : 𝑋 \to 𝑍$ .

See Quotient category

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