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{{For|the term as used in elementary geometry|congruence (geometry)}}


In [[abstract algebra]], a '''congruence relation''' (or simply '''congruence''') is an [[equivalence relation]] on an [[algebraic structure]] (such as a [[group (mathematics)|group]], [[ring (mathematics)|ring]], or [[vector space]]) that is compatible with the structure.  Every congruence relation has a corresponding [[quotient]] structure, whose elements are the [[equivalence class]]es (or '''congruence classes''') for the relation.


==Basic example==
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The prototypical example of a congruence relation is [[Modular arithmetic#Congruence relation|congruence modulo <math>n</math>]] on the set of [[integer]]s.  For a given [[positive integer]] <math>n</math>, two integers <math>a</math> and <math>b</math> are called '''congruent modulo <math>n</math>''', written
: <math>a \equiv b \pmod{n}</math>
if <math>a - b</math> is [[divisible]] by <math>n</math> (or equivalently if <math>a</math> and <math>b</math> have the same [[remainder]] when divided by <math>n</math>).
 
for example, <math>37</math> and <math>57</math> are congruent modulo <math>10</math>,
 
: <math>37 \equiv 57 \pmod{10}</math>
 
since <math>37 - 57 = -20</math> is a multiple of 10, or equivalently since both <math>37</math> and <math>57</math> have a remainder of <math>7</math> when divided by <math>10</math>.
 
Congruence modulo <math>n</math> (for a fixed <math>n</math>) is compatible with both [[addition]] and [[multiplication]] on the integers.  That is, if
 
: <math>a_1 \equiv a_2 \pmod{n}</math> and <math>b_1 \equiv b_2 \pmod{n}</math>
 
then
 
: <math>a_1 + b_1 \equiv a_2 + b_2 \pmod{n}</math>  and  <math>a_1 b_1 \equiv a_2b_2 \pmod{n}</math>
 
The corresponding addition and multiplication of equivalence classes is known as [[modular arithmetic]]. From the point of view of abstract algebra, congruence modulo <math>n</math> is a congruence relation on the [[ring (mathematics)|ring]] of integers, and arithmetic modulo <math>n</math> 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 [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[vector space]]s, [[module (mathematics)|modules]], [[semigroup]]s, [[lattice (order)|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 [[Operation (mathematics)|operations]] are [[well-defined]] on the [[equivalence classes]].
 
For example, a group is an algebraic object consisting of a [[set (mathematics)|set]] together with a single [[binary operation]], satisfying certain axioms. If <math>G</math> is a group with operation &lowast;, a '''congruence relation''' on ''G'' is an equivalence relation &equiv; on the elements of ''G'' satisfying
:''g''<sub>1</sub>&nbsp;&equiv;&nbsp;''g''<sub>2</sub>&nbsp;&nbsp;and&nbsp;&nbsp;''h''<sub>1</sub>&nbsp;&equiv;&nbsp;''h''<sub>2</sub>&nbsp;&nbsp;&nbsp;&nbsp;&rArr;&nbsp;&nbsp;&nbsp;&nbsp;''g''<sub>1</sub>&nbsp;&lowast;&nbsp;''h''<sub>1</sub>&nbsp;&equiv;&nbsp;''g''<sub>2</sub>&nbsp;&lowast;&nbsp;''h''<sub>2</sub>
for all ''g''<sub>1</sub>,&nbsp;''g''<sub>2</sub>,&nbsp;''h''<sub>1</sub>,&nbsp;''h''<sub>2</sub>&nbsp;&isin;&nbsp;''G''.  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 [[coset]]s 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
:''r''<sub>1</sub>&nbsp;+&nbsp;''s''<sub>1</sub>&nbsp;&equiv;&nbsp;''r''<sub>2</sub>&nbsp;+&nbsp;''s''<sub>2</sub>&nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp;&nbsp;''r''<sub>1</sub>''s''<sub>1</sub>&nbsp;&equiv;&nbsp;''r''<sub>2</sub>''s''<sub>2</sub>
whenever ''r''<sub>1</sub>&nbsp;&equiv;&nbsp;''r''<sub>2</sub> and ''s''<sub>1</sub>&nbsp;&equiv;&nbsp;''s''<sub>2</sub>. For a congruence on a ring, the equivalence class containing 0 is always a two-sided [[Ideal (ring theory)|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 &equiv; on an algebraic structure that satisfies
:''&mu;''(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>)&nbsp;&equiv;&nbsp;''&mu;''(''a''<sub>1</sub>&prime;, ''a''<sub>2</sub>&prime;, ..., ''a''<sub>''n''</sub>&prime;)
for every ''n''-ary operation ''&mu;'', and all elements ''a''<sub>1</sub>,...,''a''<sub>''n''</sub>,''a''<sub>1</sub>&prime;,...,''a''<sub>''n''</sub>&prime; satisfying ''a''<sub>''i''</sub>&nbsp;&equiv;&nbsp;''a''<sub>''i''</sub>&prime; for each ''i''.
 
==Relation with homomorphisms==
If &fnof;:&nbsp;''A''&nbsp;&rarr;&nbsp;''B'' is a [[homomorphism]] between two algebraic structures (such as [[group homomorphism|homomorphism of groups]], or a [[linear map]] between [[vector space]]s), then the relation &equiv; defined by
:''a''<sub>1</sub>&nbsp;&equiv;&nbsp;''a''<sub>2</sub>&nbsp;&nbsp;&nbsp;&nbsp;if and only if&nbsp;&nbsp;&nbsp;&nbsp;&fnof;(''a''<sub>1</sub>)&nbsp;=&nbsp;&fnof;(''a''<sub>2</sub>)
is a congruence relation. By the [[first isomorphism theorem]], the [[image (mathematics)|image]] of ''A'' under &fnof; is a substructure of ''B'' [[Isomorphism|isomorphic]] to the quotient of ''A'' by this congruence.
 
==Congruences of groups, and normal subgroups and ideals==
In the particular case of [[group (mathematics)|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:
#[[Given any]] element ''a'' of ''G'', ''a'' ~ ''a'' ('''[[Reflexive relation|reflexivity]]''');
#Given any elements ''a'' and ''b'' of ''G'', [[material conditional|if]] ''a'' ~ ''b'', then ''b'' ~ ''a'' ('''[[Symmetric relation|symmetry]]''');
#Given any elements ''a'', ''b'', and ''c'' of ''G'', if ''a'' ~ ''b'' [[logical conjunction|and]] ''b'' ~ ''c'', then ''a'' ~ ''c'' ('''[[Transitive relation|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''<sup>&minus;1</sup> ~ ''a' ''<sup>&minus;1</sup> (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''<sup>&minus;1</sup> * ''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 (mathematics)|ring theory]] as [[ideal (ring theory)|ideals]] instead of congruence relations, and in [[module (mathematics)|module theory]] as [[submodule]]s instead of congruence relations.
 
The most general situation where this trick is possible is with [[Omega-group]]s (in the general sense allowing operators with multiple arity). But this cannot be done with, for example, [[monoid]]s, 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'' &times; ''A'' that is both an [[equivalence relation]] on ''A'' and a [[subalgebra]] of ''A'' &times; ''A''.
 
The [[kernel (universal algebra)|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 class]]es 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 (order)|lattice]] '''Con'''(''A'') of all congruence relations on an algebra ''A'' is [[algebraic lattice|algebraic]].
 
==See also==
*[[Table of congruences]]
*[[Linear congruence theorem]]
*[[Congruence lattice problem]]
 
==References==
* Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5 discusses congruency of matrices.)
 
{{DEFAULTSORT:Congruence Relation}}
[[Category:Modular arithmetic]]
[[Category:Algebra]]
[[Category:Mathematical relations]]

Revision as of 17:54, 18 February 2014


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