In mathematics , more precisely in algebra , an equivalence relation on an algebraic structure is called a congruence relation if the fundamental operations of the algebraic structure are compatible with this equivalence relation .
Definitions
Congruence relation and quotient algebra
An equivalence relation on a set does not necessarily have anything to do with the structure that is defined on it. Particularly in algebra, however, such equivalence relations are of particular interest, their ( surjective ) quotient mapping
is compatible with the algebraic structure or a homomorphism . Because then the structure induced by on the quotient set , the so-called factor or quotient algebra from to with operations ,
-
for everyone and everyone ,
of the same kind as that of .
Such an equivalence relation is called a congruence relation on and two elements congruent after if they are equivalent with respect to :
-
.
The equivalence class of each is then called the congruence class .
An equivalence relation on is precisely then a congruence on an algebraic structure , when all the fundamental operations , are compatible with , i. H. for all , with the following applies:
-
.
Core of a homomorphism
If and are two algebraic structures of the same kind and is a homomorphism of this kind, then the kernel of
a congruence relation on and for all applies:
-
.
can be in a surjective, as follows a bijective and an injective homomorphism disassemble ( homomorphism ):
with and the inclusion image .
generalization
Quotient structure
In general, those equivalence relations on a set play an important role, their quotient mapping
with the structure in compatible or a homomorphism is.
The given structure on the quotient set , the so-called factor or quotient structure with relations ,
-
for each ,
is then again of the same kind as that of .
In particular, then all the associated functions with compatible.
Special congruences
Normal divisor of a group
Now denote a group , its neutral element and any normal subgroup of .
For each one
the associated secondary class of the normal divisor s . With
and the complex product then forms a group with the neutral element : the factor group of after .
But because
is a group homomorphism is
a congruence relation on and for all applies:
-
.
Conversely, any congruence delivers on exactly one normal subgroup in .
In a group, the normal factors correspond exactly to the congruence relations. Therefore, for any group homomorphism , the normal divisor also
becomes
referred to as the core of .
Congruence after a module
An additive Abelian group is called a module (from Latin modulus measure). Since each subgroup of a module is normal, the carrier quantities of the subgroups correspond exactly to the congruence relations on a module.
This also applies to the carrier quantities of the sub-modules of a module over a ring and in particular also to the sub-vector spaces of a vector space .
The secondary class
is designated for everyone
as a residual after or residue class modulo (from Latin. MODULO , ablative to modulus ), and the factor group 's residual module of after .
If two elements are congruent after , then they are also called congruent after the module or congruent modulo and this is written
-
or or short .
The following applies:
-
.
Is simply generated in , i.e. for a , then one also says that they are congruent modulo and notated
-
.
Examples
Identity relation
For every algebraic structure , the equivalence relation given by the graph of the identical mapping to is the equality or identity relation
-
,
a congruence relation .
All relation
To be they are equivalent any two elements. This gives an equivalence relation , the so-called universal or universal relation
-
,
it is also based on a congruence relation .
Ring ideals
Each ring is a module about itself and the carrier quantities of the associated sub- modules are exactly the ideals of the ring , therefore the ring ideals correspond exactly to the congruence relations .
L p space
In the vector space of the fold integrable functions , is
-
almost everywhere
Carrier set of a subspace of .
The quotient vector space
is called space .
Congruence of whole numbers
"Congruence" was originally called any congruence of two whole numbers modulo another whole number defined on the main ideal ring of whole numbers :
-
.
and are congruent modulo if and only if they have the same remainder when dividing by .
More congruence terms
literature
- Stanley Burris, H. P. Sankappanavar: A Course in Universal Algebra . Millennium Edition. 2012 update, ISBN 978-0-9880552-0-9 ( math.uwaterloo.ca [PDF; 4.4 MB ]).
-
Udo Hebisch , Hanns Joachim Weinert: Half rings. Algebraic Theory and Applications in Computer Science . Teubner, Stuttgart 1993, ISBN 3-519-02091-2 .
-
Thomas Ihringer : General Algebra. With an appendix on Universal Coalgebra by H. P. Gumm (= Berlin study series on mathematics . Volume 10 ). Heldermann, Lemgo 2003, ISBN 3-88538-110-9 .
- Fritz Reinhardt, Heinrich Soeder: dtv atlas for mathematics. Boards and texts . Volumes 1 and 2. 9th and 8th editions. Deutscher Taschenbuch Verlag, Munich 1991 and 1992, ISBN 3-423-03007-0 and ISBN 3-423-03008-9 .
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B. L. van der Waerden : Algebra . Using lectures by E. Artin and E. Noether. Volume I (= Heidelberg Pocket Books . Volume 12 ). 9th edition. Springer, Berlin / Heidelberg 1993, ISBN 978-3-642-85528-3 , doi : 10.1007 / 978-3-642-85527-6 .
References and comments
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↑ a b c In the literature, there is usually no clear distinction between a group and its number of carriers .