I'm wondering if there is a generalization of the Proj construction used in algebraic geometry. Given a graded ring R, which is a monoid homomorphism $R\to \mathbb{N}$, we can form the scheme Proj(R), which is the union of Spec$(R_f)_0$.

I'm wondering if there is a way to extend this construction to general homomorphisms from a ring to a (sharp) monoid.

In particular, I want to see if there is a ``Proj'' construction which gives the product $\mathbb{P}^m\times\mathbb{P}^n$ from the bidegree $k[x_0,\ldots,x_m,y_0,\ldots,y_n]\to\mathbb{N}^2$.

(I know this can be seen via two consecutive Proj, but want to see if this generalizes, as I want to see what happens if the degree map take value in a general toric monoid.)

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