For the purpose of this question, a vector addition system (VAS) is a pair (v,A) such that such that A is a finite subset of ℤᵈ for some dimension d∈ℕ₊. (Sometimes an initial vector is considered a part of the definition; saying that a VAS is a pair (v,A) of finite set A⊆ℤᵈ and v∈ℤᵈ for some d∈ℕ₊ would be a valid alternative.)
Consider the set of all such vector addition systems A (or (v,A) in the alternative formulation) such that for the dimension d of the VAS and all vectors a∈A we have ∑_{i ∈ dom a} aᵢ = 0 (here, dom a is the set of indexes of a, usually between 0 and d−1 or between 1 and d; the choice of the convention is not important). Does this set have an established name? I failed to find any myself.
Unrelated, FYI: The semantics of such a VAS A (respectively, (v,A) for the alternative formulation) is as usual: the set of walks (wⁱ)_{0≤i≤n} through componentwise nonnegative vectors (starting in w⁰ = v for the alternative formulation) such that wⁱ⁺¹ − wⁱ ∈ A for all i∈ℕ₀ satisfying i<n.
