In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.
Contents
Basic example
The prototypical example of a congruence relation is congruence modulo
if
for example,
since
Congruence modulo
if
then
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo
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
for all
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
whenever
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
for every n-ary operation μ, and all elements
Relation with homomorphisms
If
if and only if
Congruences of groups, and normal subgroups and ideals
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 operation *) and ~ is a binary relation on G, then ~ is a congruence whenever:
- Given any element a of G, a ~ a (reflexivity);
- Given any elements a and b of G, if a ~ b, then b ~ a (symmetry);
- Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c (transitivity);
- Given any elements a, a' , b, and b' of G, if a ~ a' and b ~ b' , then a * b ~ a' * b' ;
- Given any elements a and a' of G, if a ~ a' , then a−1 ~ a' −1 (this can actually be proven from the other four, so is strictly redundant).
Conditions 1, 2, and 3 say that ~ is an equivalence relation.
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 if and only if 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.
Ideals of rings and the general case
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.
A more general situation where this trick is possible is with Omega-groups (in the general sense allowing operators with multiple arity). But this cannot be done with, for example, monoids, so the study of congruence relations plays a more central role in monoid theory.
Universal algebra
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.
The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel. 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. The function that maps every element of A to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.
The lattice Con(A) of all congruence relations on an algebra A is algebraic.