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).

Modular arithmetic

The prototypical example is modular arithmetic: for n a positive natural number, two integers a and b are called congruent modulo n if a − b is divisible by n.

If a_{1}\equiv b_{1}{\pmod  n} and a_{2}\equiv b_{2}{\pmod  n}, then a_{1}+a_{2}\equiv b_{1}+b_{2}{\pmod  n} and a_{1}a_{2}\equiv b_{1}b_{2}{\pmod  n}. This turns the equivalence (mod n) into a congruence 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

P^{\top }AP=B.

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 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 PTAP = 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.

Congruences typically arise as kernels of homomorphisms, and in fact every congruence is the kernel of some homomorphism: 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. Furthermore, the function that maps every element of A to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.

Group theory

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:

  1. Given any element a of G, a ~ a;
  2. Given any elements a and b of G, if a ~ b, then b ~ a;
  3. Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c;
  4. e ~ e;
  5. Given any elements a and a' of G, if a ~ a', then a−1 ~ a'−1;
  6. Given any elements a, a', b, and b' of G, if a ~ a' and b ~ b', then a * b ~ a' * b'.

(However, we can actually shorten the list of requirements to just numbers 1, 2, 3, and 6.)

Notice that such a congruence ~ is determined entirely by the set {aG : a ~ e} of those elements of G that are congruent to the identity element, and this set is a normal subgroup. Specifically, a ~ b iff 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. This is what makes it possible to speak of kernels in group theory as subgroups, while in more general universal algebra, kernels are congruences.

Ring theory

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.

General case of kernels

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.

See also

References

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