  We study affine semigroup rings as algebras over subsemigroup rings. From
this relative viewpoint with respect to a given subsemigroup ring, the fibered
sum of two affine semigroup algebras is constructed. Such a construction is
compared to the tensor product and to the classical gluings of affine semigroup
rings as defined in Rosales (1997).
  While fibered sum can always be achieved, gluings of affine semigroup rings
do not always exist. Therefore, we further investigate when the fibered sum of
affine semigroup algebras gives rise to a gluing. A criterion is recovered in
terms of the defining semigroups under which the gluing may take place.
