Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for semigroups?
Percolation on Cayley graphs of groups is a particular case of percolation on transitive graphs. Percolation on transitive graphs satisfies many important properties which do not hold in general (see this and this papers. That provides a motivation. The Cayley graphs of semigroups are typically not transitive. Nevertheless some Cayley graphs of semigroups have been considered. For example percolation on the NE quadrant of the square lattice (which is the Cayley graph of the free commutative semigroup) is considered here. 

