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.

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    $\begingroup$ The closest thing is a convex cone. Possibly intersected with a lattice. $\endgroup$
    – lhf
    Mar 8, 2016 at 11:12
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    $\begingroup$ What is $\mathbb N_0$?? $\endgroup$
    – bof
    Mar 8, 2016 at 11:17
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    $\begingroup$ I would go with this and call it a module over the semiring $\mathbb{N}_0$ en.wikipedia.org/wiki/Module_%28mathematics%29#Generalizations $\endgroup$ Mar 8, 2016 at 11:17
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    $\begingroup$ @bof $\mathbb{N}_0 = \mathbb{N} \cup \{ 0 \}$ and it's pretty common notation, at least in Germany... $\endgroup$
    – Fryie
    Mar 8, 2016 at 11:29
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    $\begingroup$ @Fryie It is a basis of sorts, although not a vector space basis of course. When you use all he integers I've seen it called a "lattice basis" where the vectors generate a lattice of points. $\endgroup$
    – rschwieb
    Mar 8, 2016 at 11:34


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