Equivalence relation

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, ⋅) then an Equivalence relation ≡ is a congruence relation iff.

𝑔1≡𝑔2∧ℎ1≡ℎ2⟹𝑔1⋅ℎ1≡𝑔2⋅ℎ2

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 ≡ is an equivalence relation iff. 𝑓1 ≡𝑓2 :𝑋 →𝑌 and 𝑔1 ≡𝑔2 :𝑌 →𝑍 implies 𝑔1𝑓1 ≡𝑔2𝑓2 :𝑋 →𝑍.

See Quotient category

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