Finitely generated subsemigroups of $\mathbb{N}^k$

Question

It is well-know that every subsemigroup of $(\mathbb{N},+)$ is finitely generated. I am wondering if there is (any) similar characterization of subsemigroups of $\mathbb{N}^k$ for $k>1$? I am looking also for some examples of not finitely generated subsemigroups of $\mathbb{N}^k$. What are "typical" ones? Thank you for any suggestion.

Answer 1

There are indeed some non finitely generated subsemigroups of $\mathbb{N}^k$. Consider for instance the subsemigroup $S$ of $\mathbb{N}^2$ defined by $$ S = \{(m,n) \mid m > 0 \text{ and } n > 0\} $$ Then every set of generators of $S$ necessarily contains all the elements of the form $(1, n)$ with $n \geqslant 2$. Indeed, there is no way of writing these elements as the sum of several elements of $S$. It follows that $S$ is not finitely generated.