Suppose we are given a sequence of integral vectors $\alpha_1,\alpha_2,\dots,\alpha_m\in\mathbb{Z}^n$. For some $\beta\in\mathbb{Z}^n$, is there an effective algorithm to check if $$\beta\in\mathbb{N}\alpha_1+\mathbb{N}\alpha_2+\cdots+\mathbb{N}\alpha_m\ ?$$ Here $\mathbb{N}=\{0,1,2,\dots\}$.
In the linear algebra case we can use the Gaussian elimination, what about the monoid case?