The support graph $G$ is taken to be finite, simple, and connected.
Given an initial configuration $\mathbf{c}_0$, I would like to show that the set of configurations reachable from $\mathbf{c}_0$ forms a lattice under $\mathbf{a} \leq \mathbf{b}$ iff $\mathbf{b}$ is reachable from $\mathbf{a}$. (Klivans exercise 2.8.18)
Joins for convergent CFGs are not too bad, as are meets in general. I am not certain, though, about nonconvergent chip-firing games. Taking $\mathbf{c}_1$ and $\mathbf{c}_2$, it seems that there could be two possible "loops" -- if you will -- of configurations reachable corresponding to $\mathbf{c}_1$ and $\mathbf{c}_2$, but I am failing to construct such a case.
The "inverse question" -- why must two non-convergent/ infinite chip-firing sequences eventually reach the same configuration?