Congruence Relation - Definition

Definition

The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If is a group with operation ∗, a congruence relation on G is an equivalence relation ≡ on the elements of G satisfying

g₁ ≡ g₂ and h₁ ≡ h₂ ⇒ g₁ ∗ h₁ ≡ g₂ ∗ h₂

for all g₁, g₂, h₁, h₂ ∈ G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup. Together, these equivalence classes are the elements of a quotient group.

When an algebraic structure includes more than one operation, congruence relations are required to be compatible with each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on a ring must satisfy

r₁ + s₁ ≡ r₂ + s₂ and r₁s₁ ≡ r₂s₂

whenever r₁ ≡ r₂ and s₁ ≡ s₂. For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

The general notion of a congruence relation can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a congruence relation is an equivalence relation ≡ on an algebraic structure that satisfies

μ(a₁, a₂, ..., a_n) ≡ μ(a₁′, a₂′, ..., a_n′)

for every n-ary operation μ, and all elements a₁,...,a_n,a₁′,...,a_n′ satisfying a_i ≡ a_i′ for each i.