Editing Congruence relation

{{Short description|Equivalence relation in algebra}}
{{For|the term as used in elementary geometry|congruence (geometry)}}
{{no footnotes|date=February 2015}}

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 in the sense that [[algebraic operation]]s done with equivalent elements will yield equivalent elements.{{sfnp|ps=|Hungerford|1974|p=27}}  Every congruence relation has a corresponding [[Equivalence class|quotient]] structure, whose elements are the [[equivalence class]]es (or '''congruence classes''') for the relation.{{sfnp|ps=|Hungerford|1974|p=26}}

== 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 operations are [[well-defined]] on the [[equivalence class]]es.

=== General ===
The general notion of a congruence relation can be formally defined in the context of [[universal algebra]], a field which studies ideas common to all [[algebraic structures]].  In this setting, a [[binary relation|relation]] <math>R</math> on a given algebraic structure is called '''compatible''' if
: for each <math>n</math> and each <math>n</math>-ary operation <math>\mu</math> defined on the structure: whenever <math>a_1 \mathrel{R} a'_1</math> and ... and <math>a_n \mathrel{R} a'_n</math>, then <math>\mu(a_1,\ldots,a_n) \mathrel{R} \mu(a'_1,\ldots,a'_n)</math>.

A congruence relation on the structure is then defined as an equivalence relation that is also compatible.{{sfnp|ps=|Barendregt|1990|p=338|loc=Def. 3.1.1}}{{sfnp|ps=|Bergman|2011|loc=Sect. 1.5 and Exercise 1(a) in Exercise Set 1.26 (Bergman uses the expression ''having the substitution property'' for ''being compatible'')}}

== Examples ==

=== Basic example ===
{{About|the ''(mod'' n'')'' notation|the binary ''mod'' operation|modulo operation|section=yes}}

The prototypical example of a congruence relation is [[Modular arithmetic#Congruence|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]].

=== Example: Groups ===
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 <math>\ast</math>, a '''congruence relation''' on <math>G</math> is an equivalence relation <math>\equiv</math> on the elements of <math>G</math> satisfying
:<math>g_1 \equiv g_2 \ \ \,</math> and <math>\ \ \, h_1 \equiv h_2 \implies g_1 \ast h_1 \equiv g_2 \ast h_2</math> 
for all <math>g_1, g_2, h_1, h_2 \in G</math>. 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 other [[coset]]s of this subgroup.  Together, these equivalence classes are the elements of a [[quotient group]].

=== Example: Rings ===
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
: <math>r_1 + s_1 \equiv r_2 + s_2</math> and <math>r_1 s_1 \equiv r_2 s_2</math>
whenever <math>r_1 \equiv r_2</math> and <math>s_1 \equiv s_2</math>. 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.

== Relation with homomorphisms ==
If <math>f:A\, \rightarrow B</math> 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 <math>R</math> defined by
: <math>a_1\, R\, a_2</math> [[if and only if]] <math>f(a_1) = f(a_2)</math> 
is a congruence relation on <math>A</math>.  By the [[first isomorphism theorem]], the [[image (mathematics)|image]] of ''A'' under <math>f</math> is a substructure of ''B'' [[isomorphism|isomorphic]] to the quotient of ''A'' by this congruence.

On the other hand, the congruence relation <math>R</math> induces a unique homomorphism <math>f: A \rightarrow A/R</math> given by
: <math>f(x) = \{y \mid x \, R \, y\}</math>.

Thus, there is a natural correspondence between the congruences and the homomorphisms of any given algebraic structure.

== 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'', {{nowrap|''a'' ~ ''a''}} ('''[[reflexive relation|reflexivity]]''');
# Given any elements ''a'' and ''b'' of ''G'', [[material conditional|if]] {{nowrap|''a'' ~ ''b''}}, then {{nowrap|''b'' ~ ''a''}} ('''[[Symmetric relation|symmetry]]''');
# Given any elements ''a'', ''b'', and ''c'' of ''G'', if {{nowrap|''a'' ~ ''b''}} [[logical conjunction|and]] {{nowrap|''b'' ~ ''c''}}, then {{nowrap|''a'' ~ ''c''}} ('''[[Transitive relation|transitivity]]''');
# Given any elements ''a'', ''a''′, ''b'', and ''b''′ of ''G'', if {{nowrap|''a'' ~ ''a''′}} and {{nowrap|''b'' ~ ''b''′}}, then {{nowrap|''a'' * ''b'' ~ ''a''′ * ''b''′}};
# Given any elements ''a'' and ''a''′ of ''G'', if {{nowrap|''a'' ~ ''a''′}}, then {{nowrap|''a''<sup>−1</sup> ~ ''a''′<sup>−1</sup>}} (this is implied by the other four,<ref group=note>Since ''a''′<sup>−1</sup> = ''a''′<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> ~ ''a''′<sup>−1</sup> * ''a''′ * ''a''<sup>−1</sup> = ''a''<sup>−1</sup></ref> so is strictly redundant).

Conditions 1, 2, and 3 say that ~ is an [[equivalence relation]].

A congruence ~ is determined entirely by the set {{nowrap|{{mset|''a'' ∈ ''G'' | ''a'' ~ ''e''}}}} of those elements of ''G'' that are congruent to the identity element, and this set is a [[normal subgroup]].
Specifically, {{nowrap|''a'' ~ ''b''}} if and only if {{nowrap|''b''<sup>−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.

A more 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 general notion of a congruence is particularly useful in [[universal algebra]]. An equivalent formulation in this context is the following:{{sfnp|ps=|Bergman|2011|loc=Sect. 1.5 and Exercise 1(a) in Exercise Set 1.26 (Bergman uses the expression ''having the substitution property'' for ''being compatible'')}}

A congruence relation on an algebra ''A'' is a [[subset]] of the [[direct product]] {{nowrap|''A'' × ''A''}} that is both an [[equivalence relation]] on ''A'' and a [[subalgebra]] of {{nowrap|''A'' × ''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 {{nowrap|''A'' / ~}} of [[equivalence class]]es can be given the structure of an algebra in a natural fashion, the [[quotient (universal algebra)|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]].

[[John M. Howie]] described how [[semigroup]] theory illustrates congruence relations in universal algebra:
: In a group a congruence is determined if we know a single congruence class, in particular if we know the normal subgroup which is the class containing the identity. Similarly, in a ring a congruence is determined if we know the ideal which is the congruence class containing the zero. In semigroups there is no such fortunate occurrence, and we are therefore faced with the necessity of studying congruences as such. More than anything else, it is this necessity that gives semigroup theory its characteristic flavour. Semigroups are in fact the first and simplest type of algebra to which the methods of universal algebra must be applied&nbsp;...{{sfnp|ps=|Howie|1975|p=v}}

==Category theory==
In [[category theory]], a congruence relation ''R'' on a category ''C'' is given by: for each pair of objects ''X'', ''Y'' in ''C'', an equivalence relation ''R''<sub>''X'',''Y''</sub> on Hom(''X'',''Y''), such that the equivalence relations respect composition of morphisms. See {{Section link|Quotient_category#Definition}} for details.

== See also ==
* [[Chinese remainder theorem]]
* [[Congruence lattice problem]]
* [[Table of congruences]]

== Explanatory notes ==
{{reflist|group=note}}

== Notes ==
{{reflist}}

== References ==
* {{cite book | last1=Barendregt | first1=Henk | contribution=Functional Programming and Lambda Calculus |pages=321–364 |isbn=0-444-88074-7 | editor=Jan van Leeuwen | title=Formal Models and Semantics | publisher=Elsevier | series=Handbook of Theoretical Computer Science | volume=B | year=1990 }}
* {{citation |last1=Bergman |first1=Clifford |title=Universal Algebra: Fundamentals and Selected Topics |publisher=Taylor & Francis |year=2011 }}
* {{citation |last1=Horn |last2=Johnson |title=Matrix Analysis |publisher=Cambridge University Press |year=1985 |isbn=0-521-38632-2 }} (Section 4.5 discusses congruency of matrices.)
* {{citation |first1=J. M. |last1=Howie |year=1975 |title=An Introduction to Semigroup Theory |publisher=[[Academic Press]] }}
* {{citation |last1=Hungerford |first1=Thomas W. |title=Algebra |publisher=Springer-Verlag |year=1974 }}
* {{cite book |last=Rosen |first=Kenneth H |title=Discrete Mathematics and Its Applications |publisher=McGraw-Hill Education |year=2012 |isbn=978-0077418939 }}


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[[Category:Modular arithmetic]]
[[Category:Abstract algebra]]
[[Category:Binary relations]]
[[Category:Equivalence (mathematics)]]
[[Category:Universal algebra]]