Given $n > 0$ and $w \in \mathbb{Z}^n$. Is there an efficient algorithm to compute the set of irreducible elements of the monoid $M_w = \{x \in \mathbb{N}^n \mid \langle x,w\rangle = 0 \}$?
Here, $\langle x,w\rangle$ is the standard inner product, and an element $x \in M_w$ is irreducible iff $x \neq a+b$ for all non-zero $a,b \in M_w$.
This question is a cross-post from Mathematics.
