Additive relation
A submodule 
of the direct sum 
of two modules 
and 
over some ring 
. An additive relation can thus be regarded also as a (not necessary single-valued) mapping 
or, more exactly, as a "many-valued" homomorphism, i.e. a homomorphism 
of the submodule 
into the quotient module 
where
 |
 |
 |
Here, 
is the relation inverse to 
; it consists of all pairs 
such that 
. Conversely, if a submodule 
, a quotient module 
of the module 
and a homomorphism 
are given, then there also exists a unique additive relation 
such that 
.
If two additive relations 
and 
are given, then, as in the case of other binary relations, it is possible to define their product, 
, which is the set of all pairs 
such that there exists an element 
for which 
and 
. This multiplication is associative (wherever defined) and, moreover, the additive relations form a category with involution 
.
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 |
Additive relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_relation&oldid=45028