Finding spanning vector sets

Question

Let $V$ be the set of all vectors over the non-negative integers. For any two subsets $S$ and $T$ of $V$, define $S + T$ to include:

1. All vectors in $S$
2. All vectors in $T$
3. All vectors that can be expressed in the form $s + t$ where $s \in S$ and $t \in T$

Call $A$ a subset of $V$ dominant if there exists a finite subset of $V$ called $B$ with $A + B = V$. What can we say about $A$ ? How can we determine if a vector set is dominant? Is there an algorithm that produces all dominant vector sets or determines if a vector set is dominant?