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

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?

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 $$d \ge 0$$".
For any subgroup $$K \subset \mathbb{Z}^{d}$$, the monoid $$K \cap \mathbb{N}^{d}$$ is finitely generated.