I have a finite set of vectors $V\subset \mathbb{R}^n$

Let us enumerate $V = \{\tilde{v}_1, \tilde{v}_2,...,\tilde{v}_m\}$

I have some space that I want to talk about (I spend a lot of time talking about and thinking about this space): $L\subset \mathbb{R}^n$.

$$L=\{k_1\tilde{v}_1 + k_2\tilde{v}_2+...+k_m\tilde{v}_m \mid \exists k_1,k_2,...,k_m \in \mathbb{N}_0\}$$

$L$ is *almost* vector space, with "basis" $V$ and the "field" of $\mathbb{N}_0$.
But not quiet, since $\mathbb{N}_0$ is not a field, or even a ring. Among other things it does not have an inverse element of addition.

Because I spend a lot of time thinking and talking about it, I want a name for it. It seems like it should be the subject of some study, given that should show up in lots of mixed integer programming problems.