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?

I believe that Segre product, at least in the particular case of grading over $\mathbb Z$ or $\mathbb N$ (I don't know what a vector configuration is), appears in literature in many places. One such place I am aware of is the book by Polishchuk and Positselski "Quadratic Algebras", AMS, 2005, where it is used to construct nonKoszul algebras with mutually inverse Hilbert series (p. 59 onwards). It is used also in the operadic environment (see, for example, a draft of the book by Loday and Vallette at http://math.unice.fr/~brunov/Operades.html ) what encompasses the abovementioned algebra case and, probably, many other cases, including yours. 


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 Segretype product of the form $\bigoplus_i R_{id} \otimes S_{ie}$ for given bidegree $(d,e)$. This captures the idea of giving linear relations on the generating degrees (of the subalgebra of the tensor product). 

