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 in the sense that algebraic operations done with equivalent elements will yield equivalent elements.
[1] Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.
[2] 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.
The general notion of a congruence relation can be formally defined in the context of universal algebra, a field which studies ideas common to all algebraic structures.
on a given algebraic structure is called compatible if A congruence relation on the structure is then defined as an equivalence relation that is also compatible.
) is compatible with both addition and multiplication on the integers.
That is, 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
is a congruence relation on the ring of integers, and arithmetic modulo
occurs on the corresponding quotient ring.
For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms.
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 other 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 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.
is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear map between vector spaces), then the relation
is a substructure of B isomorphic to the quotient of A by this congruence.
given by Thus, there is a natural correspondence between the congruences and the homomorphisms of any given algebraic structure.
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: 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. 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.
The general notion of a congruence is particularly useful in universal algebra.
An equivalent formulation in this context is the following:[4] 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.
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.
John M. Howie described how semigroup theory illustrates congruence relations in universal algebra: In category theory, a congruence relation R on a category C is given by: for each pair of objects X, Y in C, an equivalence relation RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms.
See Quotient category § Definition for details.