For the purpose of this question, a *vector addition system* (VAS) is a pair $ (v,A)$ such that there is a dimension $ d\in\mathbb{N}_{>0}$ such that $ A$ is a finite subset of $ \mathbb{Z}^d$ and $ v\in \mathbb{N}_{\ge 0}^d$ .

Consider the set of all VAS $ (v,A)$ such that for the dimension $ d$ of $ (v,A)$ and all vectors $ a\in A$ we have $ \sum_{i<d} a_i=0$ (here, we index components by numbers ranging between 0 and $ d-1$ ). Does this set have an established name? I failed to find any myself.