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A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107001.png" /> of the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107002.png" /> of two modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107004.png" /> over some ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107005.png" />. An additive relation can thus be regarded also as a (not necessary single-valued) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107006.png" /> or, more exactly, as a  "many-valued"  homomorphism, i.e. a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107007.png" /> of the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107008.png" /> into the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a0107009.png" /> where
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070010.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070011.png" /></td> </tr></table>
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A submodule  $  r $
 +
of the direct sum  $  A \oplus B $
 +
of two modules  $  A $
 +
and  $  B $
 +
over some ring  $  R $.
 +
An additive relation can thus be regarded also as a (not necessary single-valued) mapping  $  r: A \huR B $
 +
or, more exactly, as a  "many-valued" homomorphism, i.e. a homomorphism  $  r  ^ {0} $
 +
of the submodule  $  \mathop{\rm Def}  r $
 +
into the quotient module  $  B/ \mathop{\rm Ind} (r) $
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070012.png" /></td> </tr></table>
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$$
 +
\mathop{\rm Def}  r  = \{ {a \in A } : {\exists b \in B  ( a , b )
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\in r } \}
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,
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070013.png" /> is the relation inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070014.png" />; it consists of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070016.png" />. Conversely, if a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070017.png" />, a quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070018.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070019.png" /> and a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070020.png" /> are given, then there also exists a unique additive relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070022.png" />.
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$$
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\mathop{\rm Ker}  r  = \{ a \in A : (a , 0 ) \in r \} ,
 +
$$
  
If two additive relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070024.png" /> are given, then, as in the case of other binary relations, it is possible to define their product, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070025.png" />, which is the set of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070026.png" /> such that there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070027.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070029.png" />. This multiplication is associative (wherever defined) and, moreover, the additive relations form a [[Category with involution|category with involution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010700/a01070030.png" />.
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$$
 +
\mathop{\rm Ind}  r  =   \mathop{\rm Ker}  r  ^ {-1} .
 +
$$
 +
 
 +
Here,  $  r  ^ {-1} : B \huR A $
 +
is the relation inverse to  $  r $;
 +
it consists of all pairs  $  (b, a) \in B \oplus A $
 +
such that  $  (a, b) \in r $.  
 +
Conversely, if a submodule  $  S \subset  A $,
 +
a quotient module  $  B/L $
 +
of the module  $  B $
 +
and a homomorphism  $  \beta : S \rightarrow B/L $
 +
are given, then there also exists a unique additive relation  $  r:  A \huR B $
 +
such that  $  r  ^ {0} = \beta $.
 +
 
 +
If two additive relations  $  r:  A \huR B $
 +
and  $  s:  B \huR C $
 +
are given, then, as in the case of other binary relations, it is possible to define their product, $  sr: A \huR C $,  
 +
which is the set of all pairs $  (a, c) \in A \oplus C $
 +
such that there exists an element $  b \in B $
 +
for which $  (a, b) \in r $
 +
and $  (b, c) \in s $.  
 +
This multiplication is associative (wherever defined) and, moreover, the additive relations form a [[Category with involution|category with involution]] $  r \rightarrow r  ^ {-1} $.
  
 
Additive relations are used in natural definitions of connecting homomorphisms for exact sequences of complexes. The above considerations are valid not only in the category of modules, but also in any other Abelian category.
 
Additive relations are used in natural definitions of connecting homomorphisms for exact sequences of complexes. The above considerations are valid not only in the category of modules, but also in any other Abelian category.

Revision as of 16:09, 1 April 2020


A submodule $ r $ of the direct sum $ A \oplus B $ of two modules $ A $ and $ B $ over some ring $ R $. An additive relation can thus be regarded also as a (not necessary single-valued) mapping $ r: A \huR B $ or, more exactly, as a "many-valued" homomorphism, i.e. a homomorphism $ r ^ {0} $ of the submodule $ \mathop{\rm Def} r $ into the quotient module $ B/ \mathop{\rm Ind} (r) $ where

$$ \mathop{\rm Def} r = \{ {a \in A } : {\exists b \in B ( a , b ) \in r } \} , $$

$$ \mathop{\rm Ker} r = \{ a \in A : (a , 0 ) \in r \} , $$

$$ \mathop{\rm Ind} r = \mathop{\rm Ker} r ^ {-1} . $$

Here, $ r ^ {-1} : B \huR A $ is the relation inverse to $ r $; it consists of all pairs $ (b, a) \in B \oplus A $ such that $ (a, b) \in r $. Conversely, if a submodule $ S \subset A $, a quotient module $ B/L $ of the module $ B $ and a homomorphism $ \beta : S \rightarrow B/L $ are given, then there also exists a unique additive relation $ r: A \huR B $ such that $ r ^ {0} = \beta $.

If two additive relations $ r: A \huR B $ and $ s: B \huR C $ are given, then, as in the case of other binary relations, it is possible to define their product, $ sr: A \huR C $, which is the set of all pairs $ (a, c) \in A \oplus C $ such that there exists an element $ b \in B $ for which $ (a, b) \in r $ and $ (b, c) \in s $. This multiplication is associative (wherever defined) and, moreover, the additive relations form a category with involution $ r \rightarrow r ^ {-1} $.

Additive relations are used in natural definitions of connecting homomorphisms for exact sequences of complexes. The above considerations are valid not only in the category of modules, but also in any other Abelian category.

References

[1] S. MacLane, "Homology" , Springer (1963)
[2] D. Puppe, "Korrespondenzen in Abelschen Kategorien" Math. Ann , 148 (1962) pp. 1–30
How to Cite This Entry:
Additive relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_relation&oldid=17530
This article was adapted from an original article by A.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article