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
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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