On z-Superstable and Critical Configurations of Chip Firing Pairs

It is well known that there is a duality map between the superstable configurations and the critical configurations of a graph. This was extended to all M-matrices in (Guzmàn-Klivans 2015). We show a natural way to extend this to all $(L,M)$-chip firing pairs introduced in (Guzmàn-Klivans 2016). In addition, we study various properties of this map.