In terms of elements this means that a bilinear map is a function of sets that satisfies for all elements and the two relations
Notice that this is not a group homomorphism out of the direct product group. The product group is the group whose elements are pairs with and , and whose group operation is
and hence in particular
which is (in general) different from the behaviour of a bilinear map.
The definition of tensor product of abelian groups is precisely such that the following is an equivalent definition of bilinear map:
For a function of sets is a bilinear map from and to precisely if it factors through the tensor product of abelian groups as
The analogous defintion for more than two arguments yields multilinear maps. There is a multicategory of abelian groups and multilinear maps between them; the bilinear maps are the binary morphisms, and the multilinear maps are the multimorphisms.
More generally :
As before, this is equivalent to factoring through the tensor product of modules
Multilinear maps are again a generalisation.
See at tensor product of ∞-modules
binary function, bilinear map, multilinear map