Congruence relation

In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s).

The prototypical example is modular arithmetic: If a₁ = a₂ (mod n) and b₁ = b₂ (mod n), then a₁ + b₁ = a₂ + b₂ (mod n) and a₁b₁ = a₂b₂ (mod n). This turns the equivalence (mod n) into a congruence on the ring of all integers.

The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A.

Congruences typically arise as kernels of homomorphisms, and in fact every congruence is the kernel of some homomorphism: For a given congruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra[?]. Furthermore, the function that maps every element of A to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.

In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group (with identity element e) and ~ is a binary relation on G, then ~ is a congruence whenever:

1. Given any element a of G, a ~ a;
2. Given any elements a and b of G, if a ~ b, then b ~ a;
3. Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c;
4. e ~ e;
5. Given any elements a and a' of G, if a ~ a', then a⁻¹ ~ a'⁻¹;
6. Given any elements a, a', b, and b' of G, if a ~ a' and b ~ b', then a * b ~ a' * b'.

(However, we can actually shorten the list of requirements to just numbers 1, 2, 3, and 6.)

Notice that such a congruence ~ is determined entirely by the set {a ∈ G : a ~ e} of those elements of G that are congruent to the identity element, and this set is a normal subgroup. Specifically, a ~ b iff b⁻¹ * a ~ e. So instead of talking about congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruence corresponds uniquely to some normal subgroup of G. This is what makes it possible to speak of kernels in group theory as subgroups, while in more general universal algebra, kernels are congruences.

A similar trick allows one to speak of kernels in ring theory as ideals instead of congruence relations, and in module theory as submodules[?] instead of congruence relations. The most general situation where this trick is possible is in ideal supporting algebras[?]. But this cannot be done with, for example, monoids, so the study of congruence relations plays a more central role in monoid theory.