# Name for "Almost a vector space, but with $\mathbb{N}_0$ instead of a field"

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.

• The closest thing is a convex cone. Possibly intersected with a lattice.
– lhf
Mar 8, 2016 at 11:12
• What is $\mathbb N_0$??
– bof
Mar 8, 2016 at 11:17
• I would go with this and call it a module over the semiring $\mathbb{N}_0$ en.wikipedia.org/wiki/Module_%28mathematics%29#Generalizations Mar 8, 2016 at 11:17
• @bof $\mathbb{N}_0 = \mathbb{N} \cup \{ 0 \}$ and it's pretty common notation, at least in Germany... Mar 8, 2016 at 11:29
• @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. Mar 8, 2016 at 11:34