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Congruence relation

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See congruence (geometry) for the term as used in elementary geometry.

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).

1 Modular arithmetic
2 Linear algebra
3 Universal algebra
4 Congruences of groups, and normal subgroups and ideals
- 4.1 Ideals of rings and the general case of kernels
5 See also
6 References

Modular arithmetic

The prototypical example is modular arithmetic: for n a positive integer, two integers a and b are called congruent modulo n if a − b is divisible by n (or an equivalent condition is that they give the same remainder when divided by n). For example, 5 and 11 are congruent modulo 3:

<math>11 \equiv 5 \pmod 3</math>

because 11 − 5 gives 6, which is divisible by 3. Or, equally, both numbers give the same remainder when divided by 3:

<math>11 = 3\times 3 + 2</math>

<math>5 = 1\times 3 + 2</math>

If a₁ ≡ b₁ (mod n) then a₁ + a₂ ≡ b₁ + b₂ (mod n) and a₁a₂ ≡ b₁b₂ (mod n) This turns the congruence (mod n) into an equivalence on the ring of all integers.

Linear algebra

Two real matrices A and B are called congruent if there is an invertible real matrix P such that

A symmetric matrix has real eigenvalues. The inertia of a symmetric matrix is a triple consisting of the number of positive eigenvalues, the number of zero eigenvalues, and the number of negative eigenvalues. Sylvester's law of inertia states that two symmetric real matrices are congruent if and only if they have the same inertia. So, congruence transformations may change the eigenvalues of a matrix but they cannot change the signs of the eigenvalues. For complex matrices, we have to distinguish between ^Tcongruency (A and B are ^Tcongruent if there is an invertible matrix P such that P^TAP = B) and *congruency (A and B are *congruent if there is an invertible matrix P such that P*AP = B).

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 of all congruence relations on an algebra is algebraic.

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 ~ 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 of kernels

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.