Many apologies if this is elementary, but consider the following. Let $D=\{z_1,\ldots,z_d\}$ be some set of $d$ distinct complex numbers. Is there a name for the following set: \begin{align*} S=\left\{\sum_{j=1}^{d}n_j z_j: \{n_1,\ldots,n_d\}\subset\mathbb{N}\right\} \end{align*} where we allow for $0\in\mathbb{N}$? At a minimum, it appears to be a commutative monoid isomorphic to $\mathbb{N}^d$. In particular, I'm interested in the geometry (e.g., the boundaries, when they exist) of these sets within the complex plane. For $d=2$, this generates an (non-negative) integer lattice spanned by $z_1$ and $z_2$. For $d>2$, it seems to be something like a union of (non-negative) integer lattices spanned by each size 2 subset.
I'm sure it can also be generalized for non-finite $D$ (say, countably infinite), but the finite case will do for now. It may also be interesting to consider the case where we put an upper bound on the size of the $n_j$. From playing with numerics, these sets seem to be bounded some interesting convex polygonal shapes.
Many thanks to anyone who can point me in a useful direction!