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).

Contents

1 Modular arithmetic
2 Linear algebra
3 Universal algebra
4 Group theory
5 Ring theory
6 General case of kernels
7 See also
8 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:

11 ≡ 5 (mod 3)

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

11 = 3×3 + 2

5 = 1×3 + 2

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_1a_2 \equiv b_1b_2 \pmod 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

[ P^\top A P = 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 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.

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 ~.

(See also Algebraic lattice.)

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:

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 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);
Given any elements a, a' , b, and b' of G, if a ~ a' and b ~ b' , then a * b ~ a' * b' .

Notice that ~ is trivially an equivalence relation by 1, 2 and 3.

Also notice that such a congruence ~ is determined entirely by the set 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. 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.

References

Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5 discusses congruency of matrices.)

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