Let $k$ be a field and $A\subset \mathbb{N}^d$ a vector configuration. Let $R,S$ be commutative $k$-algebras, both graded by the affine semigroup $\mathbb{N}A$. Is the 'multidgraded Segre product' $R \otimes\_{\mathbb{N} A} S := \bigoplus\_{\mathbb{N} A} R\_a \otimes\_k S\_a$ known in the literature?



Where did the multigraded Segre product appear in the literature?
I found a precursor of the notion in the paper "On unmixedness theorem" by Chow (American Journal of Mathematics, Vol. 86, No. 4, Oct., 1964). He considers a Segre-type product of the form $\bigoplus\_i R\_{id} \otimes S\_{ie}$ for given bi-degree $(d,e)$. This captures the idea of giving linear relations on the generating degrees (of the sub-algebra of the tensor product).


