Finite generation of a certain class of monoids in $\mathbb{N}^k$

Question

Is there a general classification of when linear monoids (i.e. monoids given by the solutions to a linear integral equation) are finitely generated in $\mathbb{N}^k$ or in $\mathbb{Z}^k$?

Algebraic geometry background:

The claim that quasi-projective $A$-schemes are finite type can be reframed as a statement about finite generation of certain monoids in $\mathbb{N}^k$, in the following way. We would like to show that if $S_\bullet$ is a finitely generated graded $A$-algebra, say with generators $(s_1,\ldots, s_k)$, then for any $f$ homogeneous of positive degree, $$ ((S_\bullet)_f)_0 $$ is finitely generated as an $A$-algebra.

This subring is generated over $A$ by all monomials of the form $$ \left\{\frac{s_{1}^{e_1} \cdots s_{k}^{e_k}}{f^n} : n \deg f = \sum_{j = 1}^k e_j \deg s_j\right\}. $$ The tricky algebraic fact to prove is that a finite subset of these monomials suffices to generate the rest under multiplication. Equivalently, we would like to know that the additive monoid $$ \left\{(e_1, \ldots, e_k, n) \in \mathbb{N}^{k+1} : n \deg f = \sum_{j = 1}^k e_j \deg s_j \right\} $$ is finitely generated for arbitrary positive coefficients $\deg s_i, \deg f$.

The result is true in this case (we can bound the degree of monomials needed), but does it follow from a more general fact about finite generation of linear monoids? Or is this a very special case, with exactly one coefficient of opposite sign than the others?

Answer 1

I think you may be interested in Corollary 2.11 (a) of Bruns, Gubeladze "Polytopes, rings and K-theory":

Let $M$ and $N$ be affine submonoids of $\mathbb{R}^{d}$. Then $M \cap N$ is an affine monoid.

Their definition (Definition 2.1) of an affine monoid is a monoid which is "finitely generated and isomorphic to a submonoid of a free abelian group $\mathbb{Z}^{d}$ for some $d \ge 0$".

See also Orest Bucicovschi's answer to this question which gives a simpler argument for a special case:

For any subgroup $K \subset \mathbb{Z}^{d}$, the monoid $K \cap \mathbb{N}^{d}$ is finitely generated.