Congruence relation

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. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.
Basic example.
The prototypical example of a congruence relation is congruence modulo formula_1 on the set of integers. For a given positive integer formula_1, two integers formula_3 and formula_4 are called congruent modulo formula_1, written
if formula_7 is divisible by formula_1 (or equivalently if formula_3 and formula_4 have the same remainder when divided by formula_1).
For example, formula_12 and formula_13 are congruent modulo formula_14,
since formula_16 is a multiple of 10, or equivalently since both formula_12 and formula_13 have a remainder of formula_19 when divided by formula_14.
Congruence modulo formula_1 (for a fixed formula_1) 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 formula_1 is a congruence relation on the ring of integers, and arithmetic modulo formula_1 occurs on the corresponding quotient ring.
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 formula_29 is a group with operation formula_30, a congruence relation on formula_29 is an equivalence relation formula_32 on the elements of formula_29 satisfying
for all formula_36, formula_37, formula_38, formula_39. 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
whenever formula_41. 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 formula_32 on an algebraic structure that satisfies
for every formula_1-ary operation formula_45 and all elements formula_46 such that formula_47 for each formula_48
Relation with homomorphisms.
If formula_49 is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear map between vector spaces), then the relation formula_50 defined by
is a congruence relation. By the first isomorphism theorem, the image of "A" under formula_53 is a substructure of "B" isomorphic to the quotient of "A" by this congruence.
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:
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.
John M. Howie described how semigroup theory illustrates congruence relations in universal algebra: